Content-Type: multipart/mixed; boundary="-------------0110172331587"

This is a multi-part message in MIME format.

---------------0110172331587
Content-Type: text/plain; name="01-383.keywords"
Content-Transfer-Encoding: 7bit
Content-Disposition: attachment; filename="01-383.keywords"

self-avoiding walk, Sierpinski gasket, mean square displacement, 
renormalization group
---------------0110172331587
Content-Type: application/x-tex; name="ht4dmsd.tex"
Content-Transfer-Encoding: 7bit
Content-Disposition: inline; filename="ht4dmsd.tex"

\iffalse
20000315;27--31;0401--10;13;16;17;0721;0926--30;1001--11;18;20--24;
27--31;1102--05;1205;20010221--27;0301--0304;14--22;27;0401;02;
\fi
\iftrue
\documentclass{article}
\usepackage{amssymb,latexsym}
%NG for latex 2.09
\def\le{\leqq} \def\leq{\leqq}\def\ge{\geqq} \def\geq{\geqq}
\else
\documentstyle{article}
\def\textrm#1{{\rm #1}}\def\textit#1{{\it #1}}
\def\mathrm#1{{\rm #1}}\def\mathit#1{{#1}} \def\mathbf#1{{#1}}
\def\mathbb#1{{\bf #1}}
\fi
\newcommand{\unitmatrix}{{\mathbb I}}\newcommand{\zeromatrix}{{\mathbb O}}
\newcommand{\reals}{{\mathbb R}}\newcommand{\preals}{{\mathbb R}_+}
\newcommand{\integers}{{\mathbb Z}}\newcommand{\pintegers}{{\mathbb Z}_+}
\newcommand{\nintegers}{{\mathbb N}}
\newcommand{\nreals}[1]{{\mathbb R^{#1}}}\newcommand{\cmplx}{{\mathbb C}}

\date{2001/04/02}

\setlength{\textwidth}{6.4in} \setlength{\textheight}{9.7in}
\setlength{\topmargin}{0in}
\setlength{\headheight}{0in}\setlength{\headsep}{0in}
\setlength{\oddsidemargin}{0in}\setlength{\evensidemargin}{0in}
\setlength{\footskip}{0.5in}

\newtheorem{prp}{Proposition}
\newtheorem{lem}[prp]{Lemma}\newtheorem{thm}[prp]{Theorem}
\newtheorem{cor}[prp]{Corollary}
\newenvironment{prf}{\begin{trivlist}\item[{\textit{Proof.}}]}{\end{trivlist}
  \bigskip\par}
\newenvironment{prfof}[1]{\begin{trivlist}\item[{\textit{Proof of #1.}}]}{
  \end{trivlist} \bigskip \par}
\newenvironment{rem}{\begin{trivlist}\item[{\textit{Remark.}}]}{\end{trivlist}
  \bigskip\par}
\newenvironment{xmp}{\begin{trivlist}\item[{\textit{Example.}}]}{\end{trivlist}
  \bigskip\par}
\def\QED{\relax\ifmmode\let\@tempa\relax\ifcase\@eqcnt\def\@tempa{& & &}\or
  \def\@tempa{& &}\else\def\@tempa{&}\fi\@tempa $\Box$ \else\hfill $\Box$ \fi}
\def\DDD{\relax\ifmmode\let\@tempa\relax\ifcase\@eqcnt\def\@tempa{& & &}\or
 \def\@tempa{& &}\else\def\@tempa{&}\fi\@tempa $\Diamond$
 \else\hfill $\Diamond$ \fi}
\def\prpb{\begin{prp}}\def\prpe{\end{prp}}
\def\lemb{\begin{lem}}\def\leme{\end{lem}}
\def\thmb{\begin{thm}}\def\thme{\end{thm}}
\def\corb{\begin{cor}}\def\core{\end{cor}}
\def\prfb{\begin{prf}}\def\prfe{\end{prf}}
\def\prfofb#1{\begin{prfof}{#1}}\def\prfofe{\end{prfof}}
\def\remb{\begin{rem}}\def\reme{\end{rem}}
\def\xmpb{\begin{xmp}}\def\xmpe{\end{xmp}}
\def\labelenumi{(\roman{enumi})} \def\prpi#1{(\romannumeral\ref{#1})}
\def\itmb{\begin{enumerate}}\def\itme{\end{enumerate}}
\def\itdb{\begin{itemize}}\def\itde{\end{itemize}}
\def\ittb{\begin{description}}\def\itte{\end{description}}
\def\eqnb{\begin{equation}}\def\eqne{\end{equation}}
\def\eqab{\begin{eqnarray}}\def\eqae{\end{eqnarray}}
\def\eqsb{\begin{eqnarray*}}\def\eqse{\end{eqnarray*}}
\def\arrb#1{\begin{array}{#1}}\def\arre{\end{array}}
\def\tabb#1{\par\noindent\begin{tabular}{#1}}
\def\tabe{\end{tabular}\par\noindent}
\def\eqna#1{\label{e:#1}}\def\eqnu#1{(\ref{e:#1})}
\def\prpa#1{\label{p:#1}}\def\prpu#1{Proposition~\ref{p:#1}}
\def\lema#1{\label{l:#1}}\def\lemu#1{Lemma~\ref{l:#1}}
\def\thma#1{\label{t:#1}}\def\thmu#1{Theorem~\ref{t:#1}}
\def\cora#1{\label{c:#1}}\def\coru#1{Corollary~\ref{c:#1}}
\def\seca#1{\label{s:#1}}\def\secu#1{Section~\ref{s:#1}}
\def\Rom#1{\uppercase\expandafter{\romannumeral#1}}

\def\limf#1{\displaystyle \lim_{#1\to\infty}}
\def\liminf{\displaystyle \mathop{\underline{\lim}}\limits}
\def\limsup{\displaystyle \mathop{\overline{\lim}}\limits}
\def\dsp{\displaystyle}\def\II{\sqrt{-1}}
\def\AHFA{\hbox{\vrule height16pt depth10pt width0pt}\displaystyle}
\def\prb#1{\def\prbone{#1}
  \ifx\prbone\empty{\mathrm{P}}\else{\mathrm{P[\;}}#1{\mathrm{\;]}}\fi}
\def\prbseq#1#2{\def\prbseqone{#2}
  \ifx\prbseqone\empty{\mathrm{P}}_{#1}
\else{\mathrm{P}}_{#1}{\mathrm{[\;}}#2{\mathrm{\;]}}\fi}
\def\EE#1{{\mathrm{E[\;}}#1{\mathrm{\;]}}}
\def\VV#1{{\mathrm{V[\;}}#1{\mathrm{\;]}}}

\def\sg{Sierpi\'{n}ski gasket}\def\SC{Sierpi\'{n}ski carpet}
\def\norm#1{\dsp\left\| #1 \right\|}
\def\trace{\mathop{\mathrm{tr}}\nolimits}
\def\det{\mathop{\mathrm{det}}\nolimits}
\def\diag{\mathop{\mathrm{diag}}}
\def\Dom{{\cal D}\mathit{om}}\def\Jacobi{{\cal J}}
\def\Hcomb#1#2{\setbox0=\hbox{$\displaystyle\mathrm{H}$}
 \setbox1=\hbox{$\scriptsize
 #1$}\kern \wd1{\mathrm{H}}_{\kern -1.05\wd0\kern -0.80\wd1{#1}
 \kern 1.15\wd0{#2}}}\def\neareq{
 \mbox{\setbox0=\hbox{$=$}\setbox1=\hbox{$\cdot$}\copy0\kern-\wd0%
 \raisebox{0.6\ht0}{\kern 0.2\wd0 \copy1}\kern-0.2\wd0%
 \raisebox{-0.8\ht0}{\kern 0.1\wd0 \copy1}\kern 0.4\wd0 }}
\def\binom#1#2{\biggl(\begin{array}{c} \displaystyle #1 \\
 \displaystyle #2 \end{array}\biggr)}
\def\Re{\mathbf{Re}}\def\Im{\mathbf{Im}}
\def\interior#1{{#1}^o}\def\closure#1{\bar{#1}}
\def\Borel{{\cal B}}

\makeatletter

\title   {
 Asymptotic properties of self-avoiding paths
 on $d$-dimensional {\sg} from renormalization group analysis
         }
\author  {
          Tetsuya HATTORI
          \\ {\small
          Graduate School of Mathematics, Nagoya University,
          } \\ {\small
          Chikusa-ku, Nagoya 464-8602, JAPAN
          } \\ {\small
          e-mail:
\iftrue hattori@math.nagoya-u.ac.jp
\else \verb|hattori@math.nagoya-u.ac.jp|
\fi
          } \\ {\small
          URL: http://www.math.nagoya-u.ac.jp/\~\,hattori/
          } \\
          Toshiro TSUDA
          \\ {\small
          Dai-ichi Mutual Life Insurance,
          } \\ {\small
          1--13--1 Yuraku-cho, Chiyoda-ku, Tokyo 100-8411, JAPAN
          }
         }
\begin{document}
\maketitle
%\def\thefootnote{\fnsymbol{footnote}}
%\setcounter{footnote}{1}
\def\thefootnote{\arabic{footnote}}
\setcounter{footnote}{0}

\begin{abstract}
 Notion of self avoiding fixed point, domain of attraction,
 and critical point are defined for
 self-avoiding paths on $d$-dimensional
 pre-{\sg}s, existence of which imply existence of
 limit distributions of scaled path lengths of `canonical ensemble',
 exponential growth of path numbers with respect to the length,
 and exponents for mean square displacement.

 The definitions are written in terms of the flows of the associated
  renormalization-group dynamical system.

 We apply the result to prove asymptotic behaviors of
 a certain class of self-avoiding paths on the $4$-dimensional pre-{\sg}.
\end{abstract}


%-----------------------------------------------------
\section{Introduction.}
%-----------------------------------------------------

 Self-avoiding walks on hypercubic lattices $\integers^n$
 have been studied mathematically for half a century,
 but compared to random walks (and diffusion processes,
 their continuum limits), amazingly little is known.
 For random walks,
 nice properties such as Markov properties enabled deep and accurate
 studies, many of which are effective for spaces with any dimension $n$.
 On the other hand, self-avoiding walks seem to have little such
 strong general methods. In fact, their behaviors are expected to
 vary drastically with the dimension $n$ for small $n$,
 so that effective methods possibly vary
 for different spaces.

 Turning our attention to the $2$- and $3$-dimensional {\sg}s,
 there are works on `restricted' self-avoiding walk (a subset of
 self-avoiding walks) in the physics literature \cite{Dhar},
 and mathematically rigorous studies for the full self-avoiding walk
 (including a proof that the restriced self-avoiding walk
 of \cite{Dhar} are in the
 same universality classes -- i.e., have the same exponents -- as the
 full self-avoiding walk), with further precise
 asymptotic results, exist for both the $2$-dimensional {\sg}
 \cite{HHK90,HH91,HK92} and
 the $3$-dimensional {\sg} \cite{HHK93}, but since then, no further
 development appeared, to the authors' knowledge, in the direction of
 generalizing the methods and results to arbitrary $d$-dimensional {\sg}s.

 We are therefore interested in the possibility of general theory
 of asymptotic behaviors of self-avoiding paths on
 the $d$-dimensional {\sg} ($d$SG) for all $d$.
 (It is known that self-avoiding walks on $\integers^n$
 with $n\ge 5$ have same asymptotic behavior as the corresponding simple
 random
 walks \cite{HS,HS95,MS}. The lace expansion is effective for these cases.
 However, $d$SG are, from renormalization group
 point of view, spaces `between $\integers^1$ and $\integers^2$\,'.)

 It is the purpose of this paper to formulate a generalization
 of studies on the $2$SG and $3$SG to $d$SG for all $d$.

\smallskip\par


 Previous rigorous studies for $d=2$ and $d=3$
 (and `philosophy' of renormalization group in physics)
 suggest that
 a study in self-avoiding paths on a $d$-dimensional
 {\sg} is composed of two main ingredeients:
\ittb
\item[Analysis of flows of renormalization group: ]
 To formulate the renormalization group (RG),
 a dynamical system determined by a recursion map in the scale
 parameter $n$,
 for the generating functions of the number of steps
 of the self-avoiding paths with endpoints on vertices of a simplex
 of size $2^n$, and then
 derive nice properties
 of the fixed points and trajectories of the RG flows,
 such as uniqueness of fixed point and convergence of flows.

\item[Asymptotic properties of paths: ]
 To derive asymptotic behaviors of the self-avoiding paths from
 the properties of RG flows.

\itte

 The explicit form of the RG map becomes
 rapidly complex as $d$ is increased
 (for example, see \cite[Appendix]{HHK93} for
 the explicit form of the map for $3$SG),
 and new complexities appear as $d$ is increased.
 Hence in studying asymptotic behaviors of self-avoiding paths on the $d$SG,
 one must first clarify the conditions
 for asymptotic behaviors of self-avoiding paths,
 in terms of the properties of RG flows.
 This corresponds to the latter step in the above classification.

 In the studies of random walks and diffusion processes,
 well-developed machineries such as a theory of
 Drichlet form does the work. However,
 self-avoiding walks are non-Markovian,
 and methods based on Markov properties
 do not work.
 On the other hand,
 the arguments in \cite{HHK90,HH91,HK92} relied heavily on explicit
 forms of the renormalization map for
 $d=2$ or $d=3$, and are insufficient for
 a general theory.

 In this paper, we formulate conditions
 in terms of the renormalization group (RG) flows (\secu{conditions}),
 under which, we prove that a set of self-avoiding paths
 on a $d$-dimensional pre-{\sg} has limit distribution of scaled path
 length, exponential growth of path numbers with respect to the length,
 and exponent for mean square displacement.
 \secu{definitions}, \secu{mainresults}, \secu{proofs}
 are devoted to definitions, main results, and proofs, respectively.

 We prove in \secu{deqfour} that the assumptions for
 our main results are satisfied for a certain set of self-avoiding
 paths in $d=4$\,.
 We note that the previous works for $d=2$ and $d=3$ are necessary steps
 in guiding the
 present study, as well as those for $d=4$.

\smallskip\par
{\textbf{Acknowledgements.}}
 T.~Hattori thanks Prof.~S.~Kusuoka and Prof.~K.~Hattori for collaborations.
 The research of T.~Hattori 
 is supported in part by a Grant-in-Aid for
 Scientific Research (C) from the Ministry of Education,
 Culture, Sports, Science and Technology.




%-----------------------------------------------------
\section{Renormalization group.}
%-----------------------------------------------------
\seca{definitions}

%----------------------------
\subsection{Self-avoiding paths on $d$-dimensional pre-{\sg}.}
%----------------------------

 Let $d\ge 2$ be an integer.
 We define a $d$-dimensional pre-{\sg} (pre-$d$SG) as follows.
 Consider a $d$-simplex of a unit side length
 embedded in $\reals^d$,
 and let $G_0=\{v_{0},v_{1},v_{2},\cdots,v_{d}\}$ be
 the set of vertices of the $d$-simplex, where $v_{0}=O=(0,0,\cdots,0)$
 is the origin of $\reals^d$.
 (We may occasionally also write $v_{0,i}=v_{i}$\,, $i=1,\cdots,d$.)
 Let $B_0=\{(v_{i},v_{j})\mid 0\le i<j\le d\}$ be the set of
 non-ordered pairs of vertices, and we denote the pair $(G_0,B_0)$
 by $F_0$\,.

 We define a sequence $F_n=(G_n,B_n)$\,, $n=1,2,3,\cdots$,
 of finite pre-$d$SG inductively by
\eqnb
\eqna{finitepreSG}
G_{n+1}= \bigcup_{i=0}^d (G_{n}+2^{n}v_{i}),
\ B_{n+1}=\bigcup_{i=0}^d (B_{n}+2^{n}v_{i}),
\ \ n=1,2,3,\cdots,
\eqne
 where we write $A+v=\{x+v\mid x\in A\}$ for a set $A$ and a point $v$.

 $F_{n}$ is a $d$-simplex of side length $2^{n}$, composed of $d+1$
 copies of $F_{n-1}$, with $d+1$ outmost points being
 $v_{n,0}=O$ and $v_{n,i}=2^{n} v_{i}$\,, $i=1,2,3,\cdots,d$.
 $G_{n}$ is a collection of vertices in the copies of $G_{n-1}$,
 and $B_{n}$ is a collection of bonds in the copies of $B_{n-1}$\,.

 We call
\eqnb
\eqna{preSG}
F=(G,B);
\ \ G=\bigcup_{n=0}^{\infty} G_n\,,
\ B=\bigcup_{n=0}^{\infty} B_n\,,
\eqne
 the $d$-dimensional pre-{\sg} (pre-$d$SG).
 We identify $(v,v')\in B$ with line segments $\overline{vv'}$
 whenever it would be natural to do so.


%\subsection{Self-avoiding paths on pre-\sg.}

 Denote the set of non-negative integers by $\pintegers$, and
 for $w:\ \pintegers \to G$, denote
 by $L(w)\in\pintegers\cup\{\infty\}$ (`the length of $w$')
 the smallest integer satisfying
\[ w(i)=w(L(w)),\ \ i\ge L(w). \]
 Define the set of self-avoiding paths $W_{0}$
 to be the set of maps $w:\ \pintegers \to G$, such that
\[ \arrb{ll} \dsp
w(i_{1})\ne w(i_{2}), & 0\le i_{1} < i_{2} \le L(w),
\\ \dsp
|w(i)-w(i+1)|=1, & 0 \le i \le L(w)-1,
\\ \dsp
\overline{w(i)w(i+1)} \in B, & 0 \le i \le L(w)-1\,.
\arre \]


%----------------------------
\subsection{Classification of self-avoiding paths.}
%----------------------------

 Denote by ${\cal T}_b$ the family of all the translations
 of $B_0$ that are subsets of $B$.
 Namely, ${\cal T}_b$ contains all the unit $d$-simplices
 which compose the pre-$d$SG.
 (with each simplex regarded as a collection of bonds).
 Put
\eqnb
\eqna{index} \arrb{l}\dsp
 {\cal I}_d=\{(i_1,i_2,\cdots,i_k)\in\pintegers^k \mid
k=1,2,3,\cdots,\ 0<i_1\le i_2\le\cdots \le i_k\,,
\\ \dsp
\phantom{ {\cal I}_d=(i_1,i_2,\cdots,i_k)\in\pintegers^k \mid}
\ i_1+i_2+\cdots+i_k+k\le d+1\},
\arre \eqne
 and denote the number of elements of ${\cal I}_d$ by
 $f_d=\sharp {\cal I}_d$.
%
\prpb
\prpa{dof}
 Let $w\in W_0$ and $\Delta\in{\cal T}_b$\,, and consider a set of bonds
\[ A=\{ \overline{w(i)w(i+1)}\in \Delta \mid i=0,1,2,\cdots,L(w) \}. \]
 If $A$ is not empty,
 then there exists $I=(i_1,i_2,\cdots,i_k)\in{\cal I}_d$ such that
 $A$ is congruent to
\eqnb
\eqna{dof}
 \Delta_I=\{\overline{O v_1 v_2 \cdots v_{i_1-1} v_{i_1}},
 \overline{v_{i_1+1} \cdots v_{i_1+i_2}},
 \cdots,
 \overline{v_{i_1+\cdots i_{k-1}+1} \cdots v_{i_1+\cdots i_k}}\},
\eqne
 where we used an abbreviation such as
 \[\overline{O v_1 v_2 \cdots v_{i_1-1} v_{i_1}}
=\overline{O v_1}, \overline{v_1 v_2}, \overline{v_2 v_3}, \cdots,
 \overline{v_{i_1-1} v_{i_1}}\,.\]
\prpe
\xmpb
\itdb
\item
 ${\cal I}_2=\{ (1), (2) \}$: $A\ne\emptyset$ is congruent to
 either $\{\overline{Ov_1}\}$ or $\{\overline{Ov_1v_2}\}$.
\item
 ${\cal I}_3=\{ (1), (2), (3), (1,1) \}$:
 There is a possibility that a path enters a unit tetrahedron twice,
 as $\{\overline{Ov_1},\overline{v_2v_3}\}$.
\item
 ${\cal I}_4=\{ (1), (2), (3), (4), (1,1), (1,2) \}$.
\itde
 Correspondingly, $f_2=2$, $f_3=4$, $f_4=6$.
 \iffalse $f_d=10,14,21,29,41$, $d=5,6,7,8,9$, $f_0=0$, $f_1=1$. \fi
\DDD\xmpe
\prfofb{\protect\prpu{dof}}
 If $A\ne\emptyset$, namely, if the path $w$ enters the unit $d$-simplex
 specified by $\Delta$, then $A$ is composed of one or more
 connected clusters.
 That is, $w$ may pass through $\Delta$ and may come back and reenter
 $\Delta$. Since $w$ is self-avoiding, the second passage does not
 intersect with the first one.
 Thus we can classify $A$ by the size of the connected segments.
 One may rearrange the segments in an increasing order of size, hence
 each class is determined by an increasing finite sequence of
 positive integers,
 $i_1\le i_2\le i_3\le \cdots\le i_k$ for some $k\ge 1$\,.
 The meaning of the conditions in the definition of ${\cal I}_d$
 should now be obvious.
 Since $\Delta$ is a translation of $B_0$ which is
 the set of bonds in the unit $d$-simplex $Ov_1v_2\cdots v_d$\,,
 the statement follows.
\QED\prfofe

 In analogy with \prpu{dof}
 we can classify the set of self-avoiding paths on $F_n$
 by ${\cal I}_d$\,,
 and also generalize to two or more self-avoiding paths.

 For $n\in\pintegers$ and $u,v\in G_n$, define $W^{(n,u,v)}$ by
\[ W^{(n,u,v)}=\{ w\in W_0 \mid w(0)=u,\ w(L(w))=v,
\ \ \ w(i)\in G_{n}\,,\ i\in\pintegers\}. \]
 For $n\in\pintegers$ and $I=(i_1,i_2,\cdots,i_k)\in{\cal I}_d$,
 define $W_I^{(n)}$ by
\eqnb
\eqna{SAPI}
 \arrb{l} \dsp
 W_I^{(n)} = \{ (w_1,w_2,\cdots,w_k)\in
 W^{(n,O,v_{n,i_1})}\times W^{(n,v_{n,i_1+1},v_{n,i_1+i_2+1})}\times
 W^{(n,v_{n,i_1+i_2+2},v_{n,i_1+i_2+i_3+2})}
\\ \dsp \phantom{W_I^{(n)} = }
 \times \cdots\times
 W^{(n,v_{n,i_1+i_2+\cdots+i_{k-1}+k-1},v_{n,i_1+i_2+\cdots+i_{k}+k-1})}
 \mid
\\ \dsp \phantom{W_I^{(n)} = \mid }
 \mbox{ if $i\ne j$ then $w_i$ and $w_j$ do not hit common points,
 and for each $j$ }
\\ \dsp \phantom{W_I^{(n)} = \mid }
\ \mbox{$w_j$ hits points }
 v_{n,i_1+i_2+\cdots+i_{j-1}+j-1},\; v_{n,i_1+i_2+\cdots+i_{j-1}+j},\;
 v_{n,i_1+i_2+\cdots+i_{j-1}+j+1},\;
\\ \dsp \phantom{W_I^{(n)} = \mid }
 \cdots\;
 v_{n,i_1+i_2+\cdots+i_{j-1}+i_j+j-1}, \mbox{ in this order, }
\\ \dsp \phantom{W_I^{(n)} = \mid }
\mbox{ but hits no other points in $\{ v_{n,\ell}\mid \ell=0,1,2,\cdots,d\}$}
\}. \arre
\eqne
 Obviously, $k$ is equal to the number of
 path segments that form an element in $W_I^{(n)}$\,.
\xmpb
 For $d=4$, there are $f_4=6$ types of sets $W_I^{(n)}$\,, which are
\ittb
\item[$\{(1)\}$: ]
 Set of paths from $O$ to $v_{n,1}$ which do not hit
 $v_{n,2}$, $v_{n,3}$, $v_{n,4}$\,.
\item[$\{(2)\}$: ]
 Set of paths from $O$ to $v_{n,2}$ passing through $v_{n,1}$
 which do not hit $v_{n,3}$, $v_{n,4}$\,.
\item[$\{(3)\}$: ]
 Set of paths from $O$ to $v_{n,3}$ passing through $v_{n,1}$ and
 $v_{n,2}$ in this order and avoiding $v_{n,4}$\,.
\item[$\{(4)\}$: ]
 Set of paths from $O$ to $v_{n,4}$ passing through $v_{n,1}$,
 $v_{n,2}$, and $v_{n,3}$ in this order.
\item[$\{(1,1)\}$: ]
 Set of pair of (self- and mutually-avoiding) paths,
 one from $O$ to $v_{n,1}$ and the other from $v_{n,2}$ to $v_{n,3}$
 neither hitting $v_{n,4}$\,.
\item[$\{(1,2)\}$: ]
 Set of pair of paths,
 one from $O$ to $v_{n,1}$ and
 the other from $v_{n,2}$ to $v_{n,4}$ via $v_{n,3}$\,.
\itte
\DDD\xmpe

 For $\dsp w\in \bigcup_{n\in\pintegers} \bigcup_{I\in{\cal I}_d} W_I^{(n)}$
 denote by $\hat{w}$ the set of bonds which $w$ passes. Namely,
 for $n\in\pintegers$ and $I=(i_1,i_2,\cdots,i_k)\in{\cal I}_d$\,,
 and for $w=(w_1,w_2,\cdots,w_k)\in W_I^{(n)}$\,,
\[ \hat{w}=
 \{ \overline{w_j(i)w_j(i+1)}\in B \mid i=0,1,2,\cdots,L(w_j)-1,
\ j=1,2,\cdots,k \}.\]
 Also define $S_I(w)$, $I\in{\cal I}_d$, by,
\eqnb
\eqna{S} S_I(w)= \{ \Delta \in {\cal T}_b \mid
\hat{w} \cap \Delta
\ \mbox{ is congruent to }\ \Delta_I\ \mbox{ of \eqnu{dof}} \}, \eqne
 and denote by $s_I(w)=\sharp S_I(w)$, the cardinality of $S_I(w)$.
 $s_I(w)$ is the number of unit $d$-simplices in $F$
 such that the trajectory of the path (or the paths) $w$ is congruent
 to $\Delta_I$\,.
 It is a generalized notion of the length of
 the path in the sense that
\eqnb
\eqna{sL}
 \sum_{i=1}^k L(w_i) = \sum_{J\in {\cal I}_d}
 |J| s_J(w),
\ \ w=(w_1,w_2,\cdots,w_k)\in W_I^{(n)}\,,
\ I=(i_1,\cdots,i_k)\in{\cal I}_d\,,
\ n\in\pintegers,
\eqne
 where, for $J\in{\cal I}_d$ we define $|J|$, the length of $J$, by
\eqnb
\eqna{lengthI}
|J|=j_1+\cdots +j_{\ell}\,,\ \ \mbox{ if }\ \ J=(j_1,\cdots,j_{\ell}).
\eqne






%----------------------------
\subsection{Parameter space and renormalization group.}
%----------------------------

 Assumptions of the main results  are stated
 in terms of the flows of the associated renormalization group (RG),
 which is a map (discrete-time dynamical system) in a
 parameter space of variables in the generating function of
 generalized path length $(s_J,\; J\in{\cal I}_d)$.
 The dynamical system is derived as the response
 in the parameter space to the change in $n$.
 A graphical property of $d$SG called finite ramifideness
 emphasizes that the RG is a finite dimensional
 dynamical system.

 Define the generating function
\[ \vec{X}_n=(X_{n,I},\ I\in{\cal I}_d):
\ \cmplx^{{\cal I}_d}\to\cmplx^{{\cal I}_d} \]
 of $(s_J\,,\ J\in{\cal I}_d)$ for a family of paths sets
 $(W_I^{(n)}\,,\ I\in{\cal I}_d)$, by,
\eqnb
\eqna{genfcn}
X_{n,I}(\vec{x})=
\sum_{w \in W_I^{(n)} } \prod_{J\in{\cal I}_d} x_J^{s_J(w)}\,,
\ \ \vec{x}=(x_J,\ J\in{\cal I}_d)\in \cmplx^{{\cal I}_d},\ n=0,1,2,\cdots.
\eqne
 The right hand side is a finite summation, so $X_{n,I}$ is defined on
 $\dsp \cmplx^{{\cal I}_d}$.

 The starting point of our analysis is the following.
\prpb
\prpa{RG}
 $\vec{X}_n=(X_{n,I},\ I\in{\cal I}_d)$, $n=0,1,2,\cdots$,
 satisfy the following recursion relations.
\[ \vec{X}_{0}(\vec{x})=\vec{x},\ \vec{x}\in\cmplx^{{\cal I}_d}, \]
and
\eqnb
\eqna{RG}
 \vec{X}_{n+1}=\vec{\Phi}\circ \vec{X}_n\,,
 \eqne
where
\[ \vec{\Phi}=(\Phi_I,\ I\in{\cal I}_d)=\vec{X}_1\,, \]
 is a $f_d$ dimensional vector valued function whose components
 are polynomials in $f_d$ variables with positive integer coefficients.
 In particular, $\preals^{{\cal I}_d}$ is an invariant set of $\vec{\Phi}$.

 The degree of each term in the polynomials are no less than $2$
 and no greater than $d+1$,
 and
 $\Phi_{(1)}$ contains terms $x_{(1)}{}^2$ and $x_{(1)}{}^{d+1}$\,.
\prpe
\prfb
 Let $n\in\pintegers$.
 $F_1$ is composed of $d+1$ $d$-simplices congruent to $F_0$\,.
 Similarly, $F_{n+1}$ is composed of $d+1$ $d$-simplices $F_n$\,.
 The similarity of the two compositions leads to a natural map
\[ \pi:\ W_I^{(n+1)}\to W_I^{(1)} \,,\ \ I\in{\cal I}_d. \]
 For each $X_{n+1,I}$\,, classify the summation in the right hand side
 of \eqnu{genfcn} (with $n+1$ in place of $n$)
 by ${\pi}(w)\in W_I^{(1)}$ to find
\[ \arrb{l} \dsp
X_{n+1,I}(\vec{x})=
\sum_{w\in W_I^{(n+1)}}\prod_{J\in{\cal I}_d} x_J ^{s_J(w)}
=\sum_{w'\in W_I^{(1)}} \sum_{w\in W_I^{(n+1)};\;{\pi}(w)= w'}
 \prod_{J\in{\cal I}_d} x_J^{s_J(w)}
\\ \dsp
=\sum_{w'\in W_I^{(1)}} \prod_{I'\in{\cal I}_d}\left(
\sum_{w''\in W_{I'}^{(n)}} \prod_{J\in{\cal I}_d} x_{J}^{s_{J}(w'')}
\right)^{s_{I'}(w')}
=\sum_{w'\in W_I^{(1)}}
\prod_{I'\in{\cal I}_d}(X_{n,I'}(\vec{x}))^{s_{I'}(w')}
=X_{1,I}(\vec{X}_{n}(\vec{x})).
\arre \]

 By definition \eqnu{genfcn},
 each term in $\Phi_I=X_{1,I}$ has a form
 $\dsp\prod_{J\in{\cal I}_d} x_J^{s_J(w)}$,
 hence its degree $\dsp \sum_{J\in{\cal I}_d} s_J(w)$ is,
 by definition \eqnu{S},
 the number of unit simplices in $F_1$ that a path $w$ passes through.
 This is bounded from above by the total number of unit simpices in $F_1$,
 which is $d+1$, and from below by $2$, because any two (outmost) vertices
 of $F_1$ is apart by length $2$.

 Positivity of coefficients of $X_{1,I}$ are obvious.
 Existence of terms $x_{(1)}^2$ and $x_{(1)}^{d+1}$ in
 $\Phi_{(1)}=X_{1,(1)}$ follows from the paths
 $\dsp \overline{Ov_{0,1}v_{1,1}}$ and
 $\dsp \overline{Ov_{0,d}(v_{0,d}+v_{0,d-1})(v_{0,d-1}+v_{0,d-2})\cdots
(v_{0,2}+v_{0,1})v_{1,1}}$ in $W^{(1)}_{(1)}$\,.
\QED\prfe

 Large $n$ means that the endpoints of the paths are far apart,
 hence it corresponds to large path length $L$.
 Intuitively speaking \prpu{RG} therefore
 gives a response to the change in the length scale
 of the system in consideration, the sets of self-avoiding paths,
 in terms of the parameter space of variables in
 the generating functions of $s_{I}$\,, the generalized path length.
 Global properties of the trajectories of the map $\vec{\Phi}$
 therefore is expected to give (and we will show that it does)
 large length asymptotic behaviors of self-avoiding paths on $d$SG.

 In analogy to the (mathematically misleading)
 terminology in physics literature, we call the discrete-time
 dynamical system on $\preals^{{\cal I}_d}$ defined by the map
 $\vec{\Phi}$, the renormalization group (RG).

%----------------------------
\subsection{Invariant sets.}
%----------------------------
 If there is a subset of $\preals^{{\cal I}_d}$ which is an invariant set of
 the RG map $\vec{\Phi}$, then the recursion \eqnu{genfcn} is naturally
 regarded as a recursion equation on the subset.

 For $I=(i_1,\cdots,i_k)$ and $J=(j_1,\cdots,j_{\ell})$ in ${\cal I}_d$\,,
 denote by $I\oplus J$ the rearrangement of
 $i_1,\cdots,i_k,j_1,\cdots,j_{\ell}$ in non-decreasing order,
 and define $\Xi_d\subset \preals^{{\cal I}_d}$ by
\eqnb
\eqna{Xi}
\Xi_d=\{ \vec{x}\in \preals^{{\cal I}_d}\mid
x_{I\oplus J}\le x_I x_J\ \mbox{ for all }\ I,J\in {\cal I}_d
\ \mbox{ such that }\ I\oplus J\in {\cal I}_d \}.
\eqne
\xmpb
 $\Xi_3=\{\vec{x}\in\preals^{{\cal I}_3}\mid x_{(11)}\le x_{(1)}^2\}$\,,
 $\Xi_4=\{\vec{x}\in\preals^{{\cal I}_4}\mid
 x_{(11)}\le x_{(1)}^2,\ x_{(12)}\le x_{(1)}x_{(2)} \}$\,.
\DDD\xmpe
\prpb
\prpa{Xiinv}
$\Xi_d$ is an invariant set of $\vec{\Phi}$.
\prpe
\prfb
 Let $I,J,I\oplus J\in{\cal I}_d$\,.
 Note that there is a natural one-to-one into map
 $\dsp W_{I\oplus J}^{(1)}\to W_{I}^{(1)}\times W_{J}^{(1)}$\,.
 For $\dsp w\in W_{I\oplus J}^{(1)}$\,,
 let $\dsp (w_1,w_2)\in  W_{I}^{(1)}\times W_{J}^{(1)}$
 be the corresponding pair.
 Then, for each $\Delta\in{\cal T}_b$\,,
 $\hat{w}\cap \Delta$ may be regarded as a composition of
 $\hat{w}_1\cap \Delta$ and $\hat{w}_2\cap \Delta$,
 hence if $\hat{w}\cap \Delta$ is congruent to $\Delta_K$ of \eqnu{dof}
 for some $K\in{\cal I}_d$\,,
 then there exists $K_1,K_2\in{\cal I}_d$ (allowing an emptyset)
 such that $K=K_1\oplus K_2$ and such that $\hat{w}_i\cap \Delta$, $i=1,2$,
 is congruent to $\Delta_{K_i}$, $i=1,2$, respectively.
 Note also that $\vec{x}\in\Xi_d$ implies
 $\dsp x_{K_1\oplus K_2}\le x_{K_1} x_{K_2}$\,.

 Therefore
 by definition (\prpu{RG} and \eqnu{genfcn}),
\[ \Phi_{I\oplus J}(\vec{x})=
\sum_{w \in W_{I\oplus J}^{(1)}} \prod_{K\in{\cal I}_d} x_K{}^{s_K(w)}
\le \sum_{w_1 \in W_{I}^{(1)}} \sum_{w_2 \in W_{J}^{(1)}}
 \prod_{K_1\in{\cal I}_d} x_{K_1}{}^{s_{K_1}(w_1)}
 \prod_{K_2\in{\cal I}_d} x_{K_2}{}^{s_{K_2}(w_2)}
=\Phi_I(\vec{x}) \Phi_J(\vec{x}). \]

\QED\prfe

 In the following, for ${\cal K}\subset{\cal I}_d$\,,
 we use a (somewhat irregular) notation
\eqnb
\eqna{invcone}
\preals^{\cal K}
= \{ \vec{x}\in\preals^{{\cal I}_d}\mid x_J=0,\ J\not\in{\cal K} \}
\ \subset\ \preals^{{\cal I}_d}.
\eqne
 We also write $\cmplx^{\cal K}\subset \cmplx^{{\cal I}_d}$,
 $\pintegers^{\cal K}\subset \pintegers^{{\cal I}_d}$, etc.

 Define
\eqnb
\eqna{Kres}
 {\cal K}_{res}=\{(1), (11), \cdots, (1\cdots 1)\}.
\eqne
 The indices in ${\cal K}_{res}$ correspond to those paths
 which go out of a simplex after single step passage
 each time they enter the simplex.
\prpb
\prpa{restrictedisinvariant}
 $\preals^{{\cal K}_{res}}$ is an invariant subset of $\vec{\Phi}$.
\prpe
\prfb
 This is proved by generalizing the arguments in the proof of
 \cite[Proposition 2.1 (4)(5)]{HHK93}.
\QED\prfe


%----------------------------
\subsection{Fixed point and domain of attraction.}
%----------------------------
\seca{conditions}

 Based on experiences with $d$SG for $d=2,3,4$, we define
 notions which are relevant for asymptotic behaviors of self-avoiding
 paths on $d$SG.

 Denote the Jacobi matrix of $\vec{\Phi}$ in \prpu{RG} by
 $\Jacobi=(\Jacobi_{IJ})$:
\eqnb
\eqna{Jacobi}
 \Jacobi_{IJ}(\vec{x})=\frac{\partial \Phi_I}{\partial x_J}(\vec{x}),
\ \ I,J\in {\cal I}_d\,,\ \vec{x}\in\cmplx^{{\cal I}_d}. \eqne

 We say that $\dsp\vec{x}_c$ is a
 {\textit{self-avoiding fixed point}},
 if the following hold.
\def\labelenumi{(FP\arabic{enumi})}
\itmb
\item
\label{FPbegin}  % (FP\ref{FPbegin})

\label{FP2}
 $\dsp\vec{\Phi}(\vec{x}_c)=\vec{x}_c$\,.

\item
\label{FP3}
 $\Jacobi(\vec{x}_c)$ in \eqnu{Jacobi} is diagonalizable
 by an invertible matrix.
 The eigenvalue $\lambda$ which is largest in absolute value satisfies
 $\lambda>1$ and
 all the other eigenvalues have absolute values
 strictly less than $1$\,.

 Denote by $\dsp\vec{v}_L=(v_{L,I},\ I\in{\cal I}_d)$
 a left eigenvector of $\Jacobi(\vec{x}_c)$ corresponding to $\lambda$;
\[ \sum_{I\in{\cal I}_d} v_{L,I} \Jacobi_{IJ}(\vec{x}_c)
=\lambda v_{L,J}\,,\ \ J\in{\cal I}_d\,, \]
 which we chose to have non-negative components
 (possible, thanks to Fr\"obenius' theorem).
 Then $\dsp v_{L,J}>0$, $J\in{\cal I}_d$\,.

 Similarly, denote by $\vec{v}_{R}$
 a right eigenvector corresponding to $\lambda$
 with non-negative components;
\[ \sum_{J\in{\cal I}_d}
 \Jacobi_{IJ}(\vec{x}_c)
 {v}_{R,{J}}
=\lambda {v}_{R,{I}}\,,
\ \ {I}\in{\cal I}_d\,. \]
 Then $\dsp {v}_{R,(1)}>0$\,.

\item
\label{FPirred}
 For all $I\in{\cal I}_d$ such that ${x}_{c,I}\ne 0$, there exists
 $m\in \pintegers^{{\cal I}_d}$, satisfying $m_{(1)}>0$
 and $m_J=0$ if $x_{c,J}=0$,
 such that
 there is a term
 $\dsp \prod_{J\in{\cal I}_{d}}x_J{}^{m_J}$ in $\Phi_I$\,.

\item
\label{FPsa}
 $\dsp\vec{x}_c\in\Xi_d\setminus \{\vec{0}\}$.

\label{FPend}
\itme
\def\labelenumi{(\roman{enumi})}

\smallskip\par

 We first note some simple consequences of the definitions.
\prpb
\prpa{lambdaapriori}
 Let $\vec{x}_c$ be a self-avoiding fixed point and $\lambda$ be
 as in (FP\ref{FP3}). Then $2<\lambda<d+1$.

\prpe
\prfb
 \prpu{RG} and \eqnu{Jacobi} imply
\[ 2 \Phi_I(\vec{x})<\sum_{J\in{\cal I}_d} \Jacobi_{IJ}(\vec{x}) x_J
< (d+1) \Phi_I(\vec{x}),\ \ \vec{x}\in\preals^{{\cal I}_d};\ x_{(1)}>0\,. \]
 This, with (FP\ref{FP2}) and (FP\ref{FP3}), further implies
\[ 2\sum_{I\in{\cal I}_d} v_{L,I} x_{c,I}
< \lambda \sum_{J\in{\cal I}_d} v_{L,J} x_{c,J}
< (d+1) \sum_{I\in{\cal I}_d} v_{L,I} x_{c,I}\,, \]
 which implies the statement.

\QED\prfe

\smallskip\par

 Note that if $\dsp\vec{x}_c$ is a self-avoiding fixed point
 then \prpu{RG} and (FP\ref{FP2}) imply
\eqnb
\eqna{FPn}
   \vec{X}_{n}(\vec{x}_c)=\vec{x}_c\,,\ \ n\in\pintegers,
\eqne
 For $n\in\pintegers$, denote the Jacobi matrices of
 $\vec{{X}}_n$ by ${\Jacobi}_{n}$:
\eqnb
\eqna{Jacobin}
 {\Jacobi}_{n,{I}{J}}(\vec{{x}})
=\frac{\partial {X}_{n, {I}}}{\partial {x}_{{J}}}
(\vec{{x}}),
\ \ {I},{J}\in {\cal I}_d\,,
\ \vec{{x}}\in\cmplx^{{\cal I}_d}.
\eqne
 In particular, we have, from \eqnu{Jacobi},
 $\dsp{\Jacobi}_{1}={\Jacobi}$.
 \prpu{RG} implies
\eqnb
\eqna{meanstept}
 {\Jacobi}_{n}(\vec{x})
=
 {\Jacobi}(\vec{{X}}_{n-1}(\vec{x}))\cdot
 {\Jacobi}(\vec{{X}}_{n-2}(\vec{x}))\cdot
 \cdots
 {\Jacobi}(\vec{{X}}_{1}(\vec{x}))\cdot
 {\Jacobi}(\vec{x}),
\ \ n\in\pintegers,
\eqne
 which further implies with \eqnu{FPn},
\eqnb
\eqna{meansteptfp}
 {\Jacobi}_{n}(\vec{x}_c)={\Jacobi}(\vec{x}_c)^n,
\ \ n\in\pintegers.
\eqne


 We note a relation between
 $0$ components of a fixed point and an invariant subset of $\vec{\Phi}$.
 In the known case of $d=2$ and $d=3$, the self-avoiding fixed points
 have $0$ components.
 We will write
\eqnb
\eqna{calK}
{\cal K}_{\vec{x}_c}=\{I\in{\cal I}_d\mid x_{c,I}\ne 0\},
\eqne
 and, as in \eqnu{invcone},
\[
\preals^{{\cal K}_{\vec{x}_c}}
= \{ \vec{x}\in\preals^{{\cal I}_d}
\mid x_J=0,\ J\not\in{\cal K}_{\vec{x}_c} \}\,.
\]
\prpb
\prpa{zerocomponent}
 $\dsp\preals^{{\cal K}_{\vec{x}_c}}$ is an invariant subset of $\vec{\Phi}$.
\prpe
\prfb
 If $x_{c,I}=0$ then $\dsp \Phi_I(\vec{x}_c)=x_{c,I}=0$\,.
 On the other hand $\Phi_I$ is a polynomial with positive coefficients.
 Therefore, each term in $\dsp \Phi_I(\vec{x})$ contains
 one of $x_J$'s such that $x_{c,J}=0$\,.
 In other words, each term in $\dsp \Phi_I(\vec{x})$ contains
 $x_J$ such that $J\not\in{\cal K}_{\vec{x}_c}$\,.

 Therefore, if $\dsp\vec{x}\in\preals^{{\cal K}_{\vec{x}_c}}$\,,
 then $\Phi_I(\vec{x})=0$ for those $I$ satisfying
 $x_{c,I}=0$, or equivalently, $I\not\in{{\cal K}_{\vec{x}_c}}$\,.
\QED\prfe

\remb
 For $d=2$ and $d=3$ it is known that
 the self-avoiding fixed point $\vec{x}_c$ is unique and that
 ${\cal K}_{res}={\cal K}_{\vec{x}_c}$\,.
\DDD\reme


 Let $\vec{x}_c$ be a self-avoiding fixed point.
 We say that $\dsp\vec{x}\in\preals^{{\cal I}_d}$ is
 {\textit{in the domain of attraction of $\vec{x}_c$}},
 if the following hold.
\def\labelenumi{(DA\arabic{enumi})}
\itmb
\item
\label{DAbegin}  % (DA\ref{DAbegin})
\label{DA2}
$\dsp \limf{n} \vec{X}_{n}(\vec{x})=\vec{x}_c$\,.

\item
\label{DAtech}
 If $x_{c,I}\ne 0$ then $x_I\ne 0$\,.

\label{DAend}

\itme
\def\labelenumi{(\roman{enumi})}
 We denote by $\Dom (\vec{x}_c)$ the set of
 $\dsp\vec{x}\in\preals^{{\cal I}_d}$ which are in the domain of
 attraction of $\vec{x}_c$\,.
\xmpb
 If $\vec{x}_c$ is a self-avoiding fixed point,
 then $\vec{x}_c\in\Dom(\vec{x}_c)$, i.e.,
 a self-avoiding fixed point satisfies
 (DA\ref{DAbegin}) -- (DA\ref{DAend}).

\DDD\xmpe

\medskip\par


%\subsection{Full model and restricted model.}

 Let ${\cal K}\subset {\cal I}_d$\,.
 Instead of \eqnu{SAPI}, we may consider a set of walks
 $\dsp W_{{\cal K},I}^{(n)}$ by
 restricting to those paths in $W_I^{(n)}$ which satisfy $s_J(w)=0$
 if $J\not\in{\cal K}$:
\eqnb
\eqna{WK}
 W_{{\cal K},I}^{(n)}=\{ w\in W_I^{(n)}\mid s_J(w)=0,\ J\not\in{\cal K} \}.
\eqne
 If ${\cal K}={\cal I}_d$\,, then we are dealing with the original
 (full) model;
 $\dsp W_{{\cal I}_d\,,I}^{(n)}=W_I^{(n)}$.
 We define the corresponding generating functions by
\eqnb
\eqna{genfcnK}
X_{{\cal K},n,I}(\vec{x})=
\sum_{w \in W_{{\cal K},I}^{(n)} } \prod_{J\in{\cal K}} x_J^{s_J(w)}\,,
\ \ \vec{x}=(x_J,\ J\in{\cal K})\in \cmplx^{{\cal K}},\ n=0,1,2,\cdots,
\ I\in{\cal I}_d\,.
\eqne

 If $\preals^{{\cal K}}$ is an invariant subset of $\preals^{{\cal I}_d}$,
 then $\vec{X}_{{\cal K},n}$ 
 satisfy \eqnu{RG}, with a convention that the components corresponding to
 $J\not\in{\cal K}$ are $0$\,.

 For ${\cal K}\subset {\cal I}_d$ and $\beta\in\reals$,
 define $\vec{x}_{can,\cal K}(\beta)
=(x_{can,{\cal K},I}(\beta),\ I\in{\cal I}_d)$
 by
\eqnb
\eqna{beta}
 x_{can,{\cal K},I}(\beta)= \left\{\arrb{ll} \dsp
e^{-\beta |I|}, & I\in{\cal K}, \\ 0, & I\not\in{\cal K},
 \arre \right. \eqne
 where $|I|$ is defined in \eqnu{lengthI}.
 Following the notions in statistical mechanics,
 the partition function for a set of self-avoiding paths
 specified by ${\cal K}$ is defined by
 $\vec{Z}_{{\cal K},n}=(Z_{{\cal K},n,I}\,,\; I\in{\cal I}_d)$, with
\eqnb
\eqna{ZK}
Z_{{\cal K},n,I}(\beta)
=\sum_{w \in  W_{{\cal K},I}^{(n)} } e^{-\beta L(w)},
\ \ \beta\in\reals,\ n=0,1,2,\cdots.
\eqne
 With \eqnu{sL}, we see that
\eqnb
\eqna{ZX}
Z_{{\cal K},n,I}(\beta)
= X_{{\cal K},n,I}(\vec{x}_{can,{\cal K}}(\beta)).
\eqne

 If ${\cal K}={\cal I}_d$ we also use an abbreviation
 $\vec{x}_{can}(\beta)=\vec{x}_{can,{\cal I}_d}(\beta)$ and
 $\vec{Z}_{n}(\beta)=\vec{Z}_{{\cal I}_d\,,n}(\beta)$. Hence
\eqnb
\eqna{Z}
Z_{n,I}(\beta)
=X_{n,I}(\vec{x}_{can}(\beta))
=\sum_{w \in W_{I}^{(n)} } e^{-\beta L(w)},
\ \ \beta\in\reals,\ n=0,1,2,\cdots.
\eqne

 In the following,
 the set of paths in \eqnu{WK} with ${\cal K}={\cal K}_{res}$
 will be called the restricted self-avoiding paths,
 the corresponding generating function \eqnu{genfcnK}, the generating
 function for the restricted model, and so on.


 We need the following additional definitions for \thmu{number}.
\def\labelenumi{(CS\arabic{enumi})}
\itmb
\item
\label{CSbegin}  % (CS\ref{CSbegin})
\label{CSfull}
 We say that $\beta_c\in\reals$ is a critical point
 of the full model if $\dsp \vec{x}_{can}(\beta_c)\in\Dom(\vec{x}_c)$
 for a self-avoiding fixed point $\vec{x}_c$\,.

\item
\label{CSrestrict}
 We say that $\beta_{c,res}\in\reals$ is a critical point
 of the restricted model if
 $\dsp \vec{x}_{can,{\cal K}_{res}}(\beta_{c,res})\in\Dom(\vec{x}_c)$
 for a self-avoiding fixed point $\vec{x}_c$\,.

\label{CSend}
\itme
\def\labelenumi{(\roman{enumi})}

Note that by definition \eqnu{Xi},
\eqnb
\eqna{betacinXi}
 \vec{x}_{can}(\beta_c)\in\Xi_d\,,\ \ \ \mbox{ and }\ \ \ 
 \vec{x}_{can,{\cal K}_{res}}(\beta_{c,res})\in\Xi_d\,.
\eqne




%-----------------------------------------------------
\section{Main results.}
%-----------------------------------------------------
\seca{mainresults}

 First we note the following characterization
 of a critical point $\beta_c$\,.
\thmb
\thma{freeenergy}
 If $\beta_c\in\reals$ is a critical point of the full model
 and $\vec{x}_c$ the corresponding fixed point
 (implicit in the definition (CS\ref{CSfull})),
 then for $I\in{\cal I}_d$\,,
\[ \limf{n} Z_{n,I}(\beta)= \left\{\arrb{ll}\dsp
0, & \beta>\beta_c\,,
\\ \dsp
x_{c,I}\,, & \beta=\beta_c\,.
\arre\right. \]
Moreover,
\[ \limf{n} \sum_{i=1}^d Z_{n,(i)}(\beta)= \infty,\ \ \  \beta<\beta_c\,. \]

 In particular, critical point (if exists) is unique.

 Similar result holds also for the restricted model (CS\ref{CSrestrict}).
\thme
\remb
 Since the critical point (if exists) is unique,
 there is one and only one self-avoiding fixed point that is related
 by (CS\ref{CSfull}) to the critical point.

 Though it is not trivial to prove the uniqueness of self-avoiding fixed
 point, we therefore can (and we will, in the proof of \thmu{number})
 talk about the unique self-avoiding fixed point that is related
 to the critical point, under the asumption that the critical point exists.
\DDD\reme

 For $I\in{\cal I}_d$, $n\in\pintegers$,
 and $\vec{x}\in\preals^{{\cal I}_d}$\,,
 define a probability measure $\mu_{\vec{x},n,I}$
 on the finite set
 $W_{I}^{(n)}$ by
\eqnb
\eqna{mun}
 \mu_{\vec{x},n,I}[\{w\}]
=\frac1{X_{n,I}(\vec{x})}
 \prod_{J \in {\cal I}_d}
 x_J^{s_J(w)},
\ \ w\in W_I^{(n)},\eqne
 whenever $X_{n,I}(\vec{x})\ne 0$\,.

 Note that
 if $x_{c,I}\ne 0$ and $\vec{x}\in\Dom(\vec{x}_c)$,
 then
 (DA\ref{DA2}) implies that
 $\dsp {X}_{n,{I}}(\vec{x})>0$ for
 sufficiently large $n$,
 hence if $\vec{x}\in\Dom(\vec{x}_c)$ then
 $\dsp {\mu}_{\vec{x},n,{I}}$ is well defined.
%
\thmb
\thma{scaledlength}
 Let $\vec{x}_c$ be a self-avoiding fixed point
 and $\vec{x}\in\Dom (\vec{x}_c)$. Then the following hold.
\itmb
\item
 There exists $f_d\times f_d$ matrix
 ${\Lambda}(\vec{x})$ whose elements
 are non-negative such that
\eqnb
\eqna{scaledmeansteps}
{\Lambda}(\vec{x})=\limf{n} \lambda^{-n} {\Jacobi}_{n}(\vec{x}),
\eqne
where $\Jacobi_n$ is as in \eqnu{Jacobi}.

\item
 For ${I}\in {\cal K}_{\vec{x}_c}$\,,
 the joint distribution of scaled generalized path lengths
 $(\lambda^{-n} {s}_{{J}}\,,\ {J}\in{\cal K}_{\vec{x}_c})$
 under
 $\dsp {\mu}_{\vec{x},n,{I}}$
 converges weakly to a Borel probability measure
 $\dsp p^*_{\vec{x},{I}}$
 on $\dsp\reals^{{\cal I}_d}$ as $n \to \infty$.
 Here, $\lambda$ is as in (FP\ref{FP3}).
 $\dsp p^*_{\vec{x},{I}}$ is supported on
 $\dsp\preals^{{\cal I}_d}$.

 The generating function $\dsp {\varphi}^*_{{I}}
= {\varphi}^*_{\vec{x},{I}}$, as a function of
 $({t}_{{J}},\ {J}\in{\cal K}_{\vec{x}_c})$,
 defined by
\[ {\varphi}^*_{\vec{x},{I}}(\vec{{t}})
= \int_0^{\infty} e^{\vec{{t}}\cdot \vec{{\xi}}}
 p^*_{\vec{x},{I}}[d\vec{{\xi}}],
\ \ \ \vec{{t}}\in\cmplx^{{\cal K}_{\vec{x}_c}}\,,
\]
 is an entire function in
 $\dsp\vec{{t}}$.

\item
 The set of functions
 $\dsp {\varphi}^*_{{I}}=\varphi^*_{\vec{x},I}$\,,
 ${I}\in{\cal K}_{\vec{x}_c}$\,,
 are uniquely determined by
\eqnb
\eqna{definingchrfcn}
 \arrb{l} \AHFA \dsp x_{c,{I}}\,
\frac{\partial {\varphi}^*_{{I}}}{\partial t_{{J}}}
(\vec{{0}})
= {\Lambda}_{{I}{J}}(\vec{x})\,
{x}_{{J}}\,,
\ \ \mbox{ if }\ {I},\, {J}\in {\cal K}_{\vec{x}_c}\,,
\\ \dsp x_{c,{I}}
 {\varphi}_{{I}}^*(\lambda \vec{{t}})
={\Phi}_{{I}}
(\vec{x}_c\,\vec{{\varphi}}^*(\vec{{t}})),
\ \ \vec{{t}}\in\cmplx^{{\cal K}_{\vec{x}_c}},
\ \ \mbox{ if }\ {I}\in {\cal K}_{\vec{x}_c}\,,
 \arre
 \eqne
 where we define
 $\dsp {\varphi}^*_{{J}}=0$
 for
 ${J}\not\in{\cal K}_{\vec{x}_c}$\,,
 and
 in the variable for $\vec{\Phi}$ we used an (irregular) notation
\[ (\vec{{x}}\,\vec{{\varphi}}^*(\vec{{t}}))_{{J}}
= {x}_{J}\, {\varphi}_{{J}}^*(\vec{{t}}),
\ \ {J}\in {\cal I}_d\,. \]

\item
 If $\vec{x}\in \Dom(\vec{x}_c)\cap \Xi_d$
 and $I\in{\cal K}_{\vec{x}_c}$\,,
 then
 the distribution of $\dsp\lambda^{-n} L(w)$,
 the scaled length of $\dsp w\in W^{(n)}_{I}$,
 under $\dsp \mu_{\vec{x},n,I}$ converges weakly
 to a Borel probability measure
 $\dsp\bar{p}^*_{\vec{x},I}$\,,
 which has a $C^{\infty}$ density
 $\dsp \bar{\rho}^*_{\vec{x},I}$\,.

 In particular,
 $\dsp \bar{\rho}^*_{\vec{x},{(1)}}(\xi)>0$, $\xi>0$\,. 

\itme
\thme
%
\remb
\itmb
\item
 If $x_{c,I}=0$, then the right hand sides in
 \eqnu{definingchrfcn} are $0$ because of
 \prpu{zerocomponent} and
 \eqnu{scaledmeansteps},
 hence the equations in \eqnu{definingchrfcn}
 are trivially correct for if $I\not\in{\cal K}_{\vec{x}_c}$\,.

\item
 Existence and positivity of density is used in \thmu{number}.

\itme
\DDD\reme


 We move on to the results on paths with step numbers fixed,
 instead of paths with endpoints fixed.

 We denote the self-avoiding paths starting from origin $O$ by $W^{(0)}$:
 $W^{(0)}=\{ w \in W_{0} \mid w(0)=O \}$.
 Also, we define, in analogy with \eqnu{WK},
 $W_{\cal K}^{(0)}=\{ w \in W^{(0)} \mid s_{J}(w)=0,\; J\not\in{\cal K} \}$,
 for ${\cal K}\subset {\cal I}_d$\,.

 For each $k\in\pintegers$, let
\[ N(k)=\sharp \{ w \in W^{(0)} \mid L(w)=k \} \]
 be the number of self-avoiding paths of length $k$ starting from $O$,
 and
 for ${\cal K}\subset {\cal I}_d$\,,
\[ N_{{\cal K}}(k)=\sharp \{w \in W_{\cal K}^{(0)} \mid L(w)=k \}. \]
%
\thmb
\thma{number}
 If there exists a critical point $\beta_c\in\reals$ of the full model,
 then
\[ \limf{k} \frac1k \,\log N(k) = \beta_{c}\,. \]

 Similarly,
 if there exists a critical point $\beta_{c,res}\in\reals$ of the
 restricted model,
 then $\dsp \limf{k} \frac1k \,\log N_{{\cal K}_{res}}(k) = \beta_{c,res}$\,.
\thme

 For each positive integer $k$,
 let $\tilde{P}_{k}$ be a distribution on $W^{(0)}$, defined by
\[
\tilde{P}_{k}[A]=
\frac1{N(k)} \sharp \{w\in A \mid L(w)=k \},\ \ A \subset W^{(0)}.
\]

 The next result shows the existence of the exponent
 for mean square displacement, which indicates
 (in a log ratio sense) that a typical
 self-avoiding path $w$ of length $L(w)=k$ deviates from the starting
 point by $|w(k)| \asymp k^{1/d_w}$,
 where
\eqnb
\eqna{dw}
 d_w=\frac{\log\lambda}{\log 2}\,.
\eqne
\thmb
\thma{msd}
 If there exists a critical point $\beta_c\in\reals$ of the full model,
 then
\[ \limf{k} \frac1{\log k} \log E_k[|w(k)|^{s\, d_w}]=s,
\ \ s\ge 0, \]
 where $E_k$ denotes expectation with respect to $\tilde{P}_{k}$\,,
 and $|\cdot|$ denotes the (Euclidean) length in $\reals^d\supset F$.

 Similar result holds also for the restricted model.

\thme




%-----------------------------------------------------
\section{Proofs.}
%-----------------------------------------------------
\seca{proofs}

%----------------------------
\subsection{Phase structure.}
%----------------------------

 Here we will prove \thmu{freeenergy}.

\prpb
\prpa{Phibasic}
 $\Phi_{(1)}(\vec{x})$ contains terms $x_{(i)}{}^2$\,, $1\le i\le d$,
\prpe
% Note also \prpu{RG} and \prpu{restrictedisinvariant}.
\prfb
 This is proved by a 
 similar graphical consideration as that
 in the proof of \prpu{RG} that $\Phi_{(1)}(\vec{x})$ contains
 a term $x_{(1)}{}^2$.

\QED\prfe
\prpb
\prpa{xconepos}
 If $\vec{x}_c$ is a self-avoiding fixed point, then
\itmb
\item $x_{c,(1)}>0$\,. Namely, $\dsp (1)\in{\cal K}_{\vec{x}_c}$\,.
\item $v_{R,I}>0$, $I\in {\cal K}_{\vec{x}_c}$\,,
 where $\vec{v}_R$ is as in (FP\ref{FP3}).
\itme
\prpe
\prfb
\itmb
\item
 $\vec{x}_c\ne \vec{0}$, by (FP\ref{FPsa}).
 Hence there exists $I\in{\cal I}_d$ such  that $x_{c,I}>0$\,.
 With (FP\ref{FPsa}), we further see that there exists $1\le i\le d$
 such that $x_{c,(i)}>0$\,.
 \prpu{xconepos} then implies that
 $x_{c,(1)}=\Phi_{(1)}(\vec{x}_c)\ge x_{c,(i)}^2>0$\,.

\item
 (FP\ref{FPirred}) and $I\in{\cal K}_{\vec{x}_c}$ impy
 $\dsp\Jacobi(\vec{x}_c)_{I,(1)}>0$\,.
 With (FP\ref{FP3}), this further implies
 $\dsp v_{R,I}\ge \frac1{\lambda} J(\vec{x}_c)_{I,(1)}\, v_{R,(1)}>0$\,.

\itme

\QED\prfe



 Let
\eqnb
\eqna{D}
 D=\{ \vec{x}\in\Xi_{d}\mid
 \sup_{n\in\pintegers} \max_{I\in{\cal I}_d} X_{n,I}(\vec{x})<\infty\},
\eqne
 and denote its exterior, boundary, interior in $\Xi_{d}$
 by $D^c$, $\partial D$, and $D^o$, respectively.
 (Namely, $D^c=\Xi_d\setminus \overline{D}$,
 $\partial D=\overline{D}\cap \overline{D^c}$,
 $D^o=D\setminus \partial D$.)
 Let also
\[ \tilde{D}=\{ \vec{x}\in\Xi_{d}\mid
\limf{n} \max_{I\in{\cal I}_d} X_{n,I}(\vec{x})=0 \}. \]
\thmb
\thma{D}
\itmb
\item
 It holds that
\eqnb
\eqna{Dleqone}
 D=\{ \vec{x}\in\Xi_{d}\mid
 \sup_{n\in\pintegers} \max_{I\in{\cal I}_d} X_{n,I}(\vec{x})\le 1\}.
\eqne
 In particular,
 $D$ is a closed subset of $\Xi_{d}$.

\item
 Let $\vec{x}\in D$ and $\vec{x}{}'\in\Xi_d$\,.
 If, for each $I\in{\cal I}_d$ either $x'_I<x_I$ or $x'_I=x_I=0$ holds,
 then $\vec{x}{}'\in \tilde{D}$.

\item It holds that
\eqnb
\eqna{Dotozero}
 D^o=\tilde{D}
\eqne

\item
 $D^c$, $\partial D$, and $D^o$ are non-empty invariant sets of $\vec{\Phi}$.

\itme
\thme
\remb
 Note that this theorem holds independently
 of the notion of self-avoiding fixed points and critical points.
\DDD\reme
\prfb
\itmb
\item
 Denote the right hand side of \eqnu{Dleqone} by $D'$.
 Obviously $D'\subset D$. To prove $D'{}^c\subset D^c$ (in $\Xi_d$),
 let $\vec{x}\in D'{}^c\subset \Xi_d$.
 Then the definition of $D'$ implies that
 there exists $n\in\pintegers$ and $I\in{\cal I}_d$ such that
 $X_{n,I}(\vec{x})>1$, and the definition of $\Xi_d$ therefore implies
 $X_{n,(i)}(\vec{x})>1$ for some $1\le i\le d$.
 Then \prpu{Phibasic} implies $X_{n+1,(1)}(\vec{x})>1$, which, with
 \prpu{RG}, further implies $\limf{n} X_{n,(1)}(\vec{x})=\infty$.
 Hence $D'{}^c\subset D^c$.

\item
 The case $\vec{x}{}'=\vec{0}$ is trivial.
 For $\vec{x}$ and $\vec{x}{}'$ in $\preals^{{\cal I}_d}\setminus\{0\}$,
 let $\dsp r=\max_{I\in{\cal I}_d;\; x_I\ne 0} \frac{x'_I}{x_I}$\,.
 Since $\Phi_I$, $I\in{\cal I}_d$\,,
 are polynomials with positive coefficients
 and each term of degree no less than $2$ (\prpu{RG}),
 we have
 $\dsp \max_{I\in{\cal I}_d} \Phi_I(\vec{x}{}')
\le r^2\max_{I\in{\cal I}_d} \Phi_I(\vec{x})$ if $0\le r\le 1$\,.
 By induction, 
 $\dsp \max_{I\in{\cal I}_d} X_{n,I}(\vec{x}{}')
\le r^{2^n} \max_{I\in{\cal I}_d} X_{n,I}(\vec{x})$, $n\in\pintegers$.
 If $\vec{x}\in D\setminus\{\vec{0}\}$ and $\vec{x}{}'\in\Xi_d$\,, and
 if for each $I\in{\cal I}_d$ either $x'_I<x_I$ or $x'_I=x_I=0$ holds,
 then $0\le r<1$\,.
 Since $\{X_{n,I}(\vec{x})\}$ is bounded,
 we have $\vec{x}{}'\in \tilde{D}$.

\item
 For $\epsilon>0$ let
 $\dsp D_{\epsilon}
=\{\vec{x}\in\Xi_d\mid \sum_{I\in{\cal I}_d} x_{I}<\epsilon\}$.
 Since  $\Phi_I$ is a polynomial with each term of order no less than $2$
 (\prpu{RG}), there exists a constant $M>0$ such that
 \[\sum_{I\in {\cal I}_d} \Phi_I(\vec{x})
\le M \epsilon \sum_{I\in {\cal I}_d} x_I\,,
 \ \ \vec{x}\in D_{\epsilon},\ 0<\epsilon<1\,. \]

 Let $\epsilon=\frac1{2M} \wedge \frac12$\,.
 Then by induction in $n$ with
\[ \sum_{I\in{\cal I}_d} X_{n+1,I}(\vec{x})
=\sum_{I\in{\cal I}_d} \Phi_I(\vec{X}_n(\vec{x})), \]
 we have, for $\vec{x}\in D_{\epsilon}$\,,
 $\vec{X}_n(\vec{x})\in D_{\epsilon}$, $n\in\pintegers$.
 Then we further have
\[ \sum_{I\in{\cal I}_d} X_{n+1,I}(\vec{x})
\le M\epsilon\sum_{I\in{\cal I}_d} X_{n,I}(\vec{x})
\le \frac12 \sum_{I\in{\cal I}_d} X_{n,I}(\vec{x}), \ \ n\in\pintegers, \]
 which implies $\vec{x}\in\tilde{D}$.
 Hence $D_{\epsilon}\subset \tilde{D}$ for
 $\epsilon=\frac1{2M} \wedge \frac12$\,.
 On the other hand, the definition of $\tilde{D}$ implies that
 if $\vec{x}\in\tilde{D}$ then $\vec{X}_n(\vec{x})\in D_{\epsilon}$
 for sufficiently large $n$. $\vec{X}_n$ is a continuous function and
 $D_{\epsilon}$ is an open set, hence there exists an open neighborhood
 $U$ of $\vec{x}$ such that $\vec{X}_n(U)\subset D_{\epsilon}$.
 Hence $\vec{X}_n(U)\subset\tilde{D}$.
 This implies that $\tilde{D}$ is an open subset of $D$.
 Since $D^o$ is the largest open set of $D$, this impleis
 $\tilde{D}\subset D^o$.

 To prove $\tilde{D}\supset D^o$, assume $\vec{x}{}'\in D^o$.
 Since $D^o$ is an open set, there exists $\vec{x}\in D^o$ such that
 $x'_I<x_I$, $I\in{\cal I}_d$\,.
 Then, the above claim imples $\vec{x}{}'\in \tilde{D}$.

\item

 By \eqnu{Dotozero} and \eqnu{Dleqone}, we see that $D$ and $D^o$ are
 invariant sets of $\vec{\Phi}$. Since $D=\overline{D}$, $D^c$ is also
 invariant, hence $\partial D$ also is.
 Obviously, $\vec{x}$ with small enough components is in $D^o$
 and that with sufficiently large components is in $D^c$,
 hence the boundary $\partial D$ is also non-empty.

\itme
\QED\prfe

\prfofb{\protect\thmu{freeenergy}}
 The case $\beta=\beta_c$ holds by the definition (CS\ref{CSfull}).

 By the definitions (CS\ref{CSfull}) and (FP\ref{FPsa}),
 $\limf{n} \vec{X}_n(\vec{x}_{can}(\beta_c))=\vec{x}_c\ne \vec{0}$\,,
 which, with \eqnu{betacinXi},
 implies $\vec{x}_{can}(\beta_c)\in \partial D$.
 Monotonicity property in \thmu{D} then implies
 $\vec{x}_{can}(\beta)\in \tilde{D}$ if $\beta>\beta_c$\,,
 hence, in particular, $\limf{n} \vec{Z}_n(\beta)=0$\,.

 Finally, if $\beta<\beta_c$ and $\vec{x}_{can}(\beta)\in D=\overline{D}$,
 then the monotonicity property in \thmu{D} implies
 $\vec{x}_{can}(\beta_c)\in \tilde{D}=D^o$, which  contradicts
 $\vec{x}_{can}(\beta_c)\in \partial D$.
 Hence $\vec{x}_{can}(\beta)\in D^c$ and
 in particular, with \eqnu{Xi},
 we have $\dsp \limf{n} \sum_{i=1}^d Z_{n,(i)}(\beta)=\infty$\,.

 The case of restricted model is similarly proved,
 if we note (CS\ref{CSrestrict}) in place of (CS\ref{CSfull}).

\QED\prfofe


 We will use the following in the proof of \thmu{number}.
\prpb
\prpa{lowtempphasefreeenergy}
 If $\vec{x}\in D^o$ then
 $\dsp \limsup_{n\to\infty}\max_{I\in{\cal I}_d}
 2^{-n}\log X_{n,I}(\vec{x}) < 0$\,.
\prpe
\prfb
 Write $\dsp Y_n=\max_{I\in{\cal I}_d} \log X_{n,I}$\,, $n\in\pintegers$.
 Since $\Phi_I$ are polynomials of positive coefficients with each term
 of degree no less than $2$ (\prpu{RG}),
 there exists a polynomial $P_1$ with positive coefficients such that
\[ Y_{n+1}(\vec{x})
\le  2 Y_{n}(\vec{x})+  \log P_1( \exp(Y_{n}(\vec{x}))),
\ \ n\in\pintegers,\ \vec{x}\in\preals^{{\cal I}_d}. \]

 If $\vec{x}\in D^o$,
 \thmu{D} implies that $\vec{X}_{n}(\vec{x})\in D^o$, $n\in\pintegers$.
 Hence, $\{\vec{X}_{n}(\vec{x})\mid n\in\pintegers\}$ is bounded,
 and there exists $M\in\reals$, a constant in $n$, such that
\eqnb
\eqna{lowtempphasefreeenergy}
 Y_{n+1}(\vec{x})\le  2 Y_{n}(\vec{x}) +M,\ \ n\in\pintegers.
\eqne
 In particular, $2^{-n} (Y_{n}(\vec{x}) +M)$ is decreasing, hence
\[ \limsup_{n\to\infty} 2^{-n} (Y_{n}(\vec{x}) +M)
\le 2^{-n_0} (Y_{n_0}(\vec{x}) +M),\ \ \ n_0\in\pintegers. \]
 Also $\vec{x}\in D^o$ implies
\[\limf{n} Y_n(\vec{x})=-\infty.\]
 Hence
\[ \limsup_{n\to\infty} 2^{-n} Y_{n}(\vec{x})
=\limsup_{n\to\infty} 2^{-n} (Y_{n}(\vec{x})+M)<0, \]
 which implies the statement.

\QED\prfe



%----------------------------
\subsection{Distribution of path length.}
%----------------------------

 Here we will prove \thmu{scaledlength}.
\prpb
\prpa{scaledmeansteps}
 Let $\vec{x}_c$ be a self-avoiding fixed point and
 $\vec{x}\in\Dom(\vec{x}_c)$ then
 there exists ${f}_d\times {f}_d$ matrix
 ${\Lambda}(\vec{x})$
 such that \eqnu{scaledmeansteps} holds.
 The elements of the matrix are non-negative, and
\eqnb
\eqna{Lambdapos}
{\Lambda}(\vec{x})_{I,(1)}>0,\ \ I\in{\cal K}_{\vec{x}_c}\,.
\eqne
\prpe
\prfb
 First note that
 (FP\ref{FP3}) and \eqnu{meansteptfp} imply an existence of the limit
 in \eqnu{scaledmeansteps} for $\vec{x}=\vec{x}_c$\,.

 Next recall the following two results.
\lemb[{\protect\cite[Lemma 4.2]{HHK93}, \cite[Lemma (3.1)]{HHK90}}]
\lema{scaledmeansteps}
 Let $N$ be a positive integer, and $A$ and $A_n$, $n\in\nintegers$,
 be $N\times N$ matrices.
 Assume that there is an invertible $N\times N$ matrix $Q$ such that
 $Q^{-1}AQ$ is a diagonal matrix whose elements are non-negative,
 and whose eigenvalue $\lambda$ that is largest in absolute value,
 is positive.
 Assume further that $\dsp \sum_{n=1}^{\infty} \norm{A_n-A}<\infty$.
 Then there exists a matrix $R$ such that
\[ \limf{m} \limsup_{n\to\infty} \norm{\lambda^{-n} A_{n+m}A_{n+m-1}
\cdots A_{m+1}\, -\, R}=0\,. \]
 $Q^{-1}RQ$ is a diagonal matrix whose diagonal elements satisfy
 $(Q^{-1}RQ)_{ii}=1$ if $(Q^{-1}AQ)_{ii}=\lambda$, and
 $(Q^{-1}RQ)_{ii}=0$ otherwise.
 Moreover $\dsp \limf{n} \lambda^{-n} A_nA_{n-1}\cdots A_1$ exists.

 ($\norm{A}$ is the standard operator norm defined as
 the square root of largest eigenvalue of $A^*A$.)

\leme
%
\lemb[{\protect\cite[Lemma 3.8]{HHK93}}]
\lema{criticaltrajectory}
 If $\vec{x}_c$ is a self-avoiding fixed point and
 $\vec{x}\in\Dom(\vec{x}_c)$, then
 there exist positive constants $\rho$ and $C$, satisfying $\rho<1$,
 such that
\[ |X_{n,I}(\vec{x})-x_{c,I}|
\le C\rho^n,\ \ n\in\pintegers,\ I\in{\cal I}_d\,. \]

\leme
%
\remb
\itmb
\item
 The proof in \cite[Proposition 3.8]{HHK93} is for $d=3$,
 but the proof is directly applicable to general cases.
 In fact, the proof is an application
 of a general theory for diffeomorphisms \cite{Irwin}.

\item
 Though we do not use this,
 the proof in \cite[Proposition 3.8]{HHK93} implies
 that \lemu{criticaltrajectory} holds for any $\rho$ satisfying
 \[ \max\{\lambda^{-1}, |\lambda_2|\}<\rho<1, \]
 where $\lambda_2$ is
 the eigenvalue of $\dsp\Jacobi(\vec{x}_c)$ which is
 second largest in absolute value.

\itme
\DDD\reme

 Let us proceed with the proof of existence of the limit
 in \prpu{scaledmeansteps}.
 By the mean-value theorem,
\eqnb
\eqna{meanvalue}
 {\Jacobi}_{{I}{J}}(\vec{x})
- {\Jacobi}_{{I}{J}}(\vec{x}_c)
= \sum_{K\in{\cal I}_d}
 \frac{\partial {\Jacobi}_{{I}{J}} }{\partial x_{K}}(\vec{u})
\, (x_{K}-x_{c,K}),
\ \ \ {I},{J}\in{\cal I}_d\,.
\eqne
 where $\dsp \vec{u}=\vec{x}_c+(\vec{x}-\vec{x}_c) \theta$
 for some $\theta\in (0,1)$.

 Since $\vec{x}\in\Dom(\vec{x}_c)$, $\vec{X}_m(\vec{x})$ is
 in a neghborhood of $\vec{x}_c$ for sufficiently large $m$.
 Therefore  $\vec{X}_m(\vec{x})$ is in a bounded domain.
 In particular,
 there exists a positive constant $M$ such that
\[ 
 \left|\frac{\partial
 {\Jacobi}_{{I}{J}} }{
\partial x_{K}}(\vec{u})\right|\le M,
\ \ \ I,J,K\in{\cal I}_d\,,
\ \vec{u}=\vec{x}_c+(\vec{X}_m(\vec{x})-\vec{x}_c) \theta,
\ 0<\theta<1\,,\ m\in\nintegers.
\]
 This with \eqnu{meanvalue} and \lemu{criticaltrajectory} implies
\[ \norm{{\Jacobi}(\vec{X}_m(\vec{x}))
- {\Jacobi}(\vec{x}_c)}
\le C'\rho^m,\ m\in\pintegers, \]
 for some  $C'>0$ and $0<\rho<1$\,.
 Therefore we see that the assumptions in \lemu{scaledmeansteps} are
 satisfied with
\eqnb
\eqna{AJacobi}
 A_n= {\Jacobi}(\vec{X}_n(\vec{x}))
\eqne
 and
 $\dsp A= {\Jacobi}(\vec{x}_c)$.
 Hence, the limit \eqnu{scaledmeansteps} exists.

 To prove \eqnu{Lambdapos},
 let $i$ be an index satisfying $(Q^{-1}AQ)_{ii}=\lambda$.
 Then the collum $i$ of the matrix $Q$
 in \lemu{scaledmeansteps}, applied to the present case,
 can be chosen to be proportional to the right eigenvector
 $\vec{{v}}_{R}$ of
 $\dsp A= {\Jacobi}(\vec{x}_c)$
 in (FP\ref{FP3}).
 Similarly, the row $i$ of matrix $Q^{-1}$,
 applied to the present case,
 can be chosen to be the left eigenvector
 $\vec{v}_L$ in (FP\ref{FP3}).
 Therefore \lemu{scaledmeansteps}
 implies that if $v_{R,I}>0$ then
 for sufficiently large $m$, 
 $\dsp\lim_{n\to\infty} \lambda^{-n} A_{n+m}A_{n+m-1}\cdots A_{m+1}$
 for $A_n$ given by \eqnu{AJacobi} has positive $(I,J)$ element
 for all $J\in{\cal I}_d$\,, in particular, for $J=(1)$.
 Also, since $\Phi_{(1)}(\vec{x})$ contains a term $x_{(1)}{}^2$
 (\prpu{RG}),
 $\dsp A_n$
 has positive
 $((1),(1))$ element, which further implies that
 $ A_{m}A_{m-1}\cdots A_{1}$ has positive $((1),(1))$ element.
 Finally, we know from \prpu{xconepos} that 
 $v_{R,I}>0$ for $I\in{\cal K}_{\vec{x}_c}$\,.
 Hence \eqnu{Lambdapos} holds.

\QED\prfe


 For $\dsp \vec{{x}}
=({x}_{{I}},\; {I}\in{\cal I}_d)
\in\cmplx^{{\cal I}_d}$
 and $\dsp \vec{{t}}
=({t}_{{I}},\; {I}\in{\cal I}_d)
\in\cmplx^{{\cal I}_d}$,
 we use an (irregular) notation
\eqnb
\eqna{xt}
 \vec{{x}}(\vec{{t}})
=({x}_{{I}} \exp(\lambda^{-n} {t}_{{I}}),
\ {I}\in{\cal I}_d).
\eqne
\prpb
\prpa{chrfcnconv}
 If $\dsp \vec{x}\in \Dom(\vec{x}_c)$
 then
 $\dsp \vec{{X}}_n(\vec{x}(\vec{t}))$
 converges uniformly in $\dsp\vec{{t}}\in\cmplx^{{\cal I}_d}$
 for any compact subset of $\dsp\cmplx^{{\cal I}_d}$.
 Furthermore,
\itmb
\item
 If $\dsp I\in{\cal I}_d$ satisfies $\dsp x_{c,I}=0$
 (i.e., if $\dsp I\not\in {\cal K}_{\vec{x}_c}$),
 then
\[
 \limf{n} {X}_{n,{I}}(\vec{{x}}(\vec{{t}}))
=0,
\ \ \vec{{t}}\in\cmplx^{{\cal I}_d}.
\]

\item
 Define $\vec{\varphi}^*=\vec{\varphi}^*_{\vec{x}}
=(\varphi^*_{\vec{x},{I}}\,,\ I\in{\cal I}_d)$
 by
\[
 {\varphi}_{\vec{x},I}^*(\vec{{t}}) =
\left\{\arrb{ll}
\frac1{\dsp x_{c,I}}
 \limf{n} {X}_{n,{I}}(\vec{{x}}(\vec{{t}})),
\ \ \vec{{t}}\in\cmplx^{{\cal I}_d},
& \mbox{ if } I\in {\cal K}_{\vec{x}_c}\,,
 \\ \dsp
0,
& \mbox{ if } I\not\in {\cal K}_{\vec{x}_c}\,.
\arre \right.
\]
 Then $\dsp \vec{\varphi}^*$ is entire and satisfies
\[ x_{c,I} {\varphi}_{{I}}^*(\lambda \vec{{t}})
={\Phi}_{{I}}(\vec{x}_c\vec{{\varphi}}^*(\vec{{t}})),
\ \ \vec{{t}}\in\cmplx^{{\cal I}_d},\ I\in{\cal I}_d\,. \]
 where, the notation of the variable for $\Phi_I$
 is as in \thmu{scaledlength}.
 It also holds that
\[ x_{c,I}\,
\frac{\partial {\varphi}^*_{{I}}}{\partial t_{{J}}}
(\vec{{0}})
={x}_{{J}}\,
 {\Lambda}_{{I}{J}}(\vec{{x}}),
\ \ I,J\in {\cal I}_d\,, \]
 where ${\Lambda}$ is defined in \eqnu{scaledmeansteps}.

\itme

\prpe
\prfb
 For $\dsp \vec{{z}}\in\cmplx^{{\cal I}_d}$, define
\[ |\vec{{z}}|_*=\sum_{I\in{\cal I}_d} v_{L,I}\, |z_I|, \]
 where $v_{L,I}$ is as in (FP\ref{FP3}).
 It is easy to see that (FP\ref{FP3}) implies that
 $|\cdot|_*$ is a norm.
 (Note that $v_{L,I}>0$\,, $I\in{\cal I}_d$\,, implies uniqueness,
 i.e., that $|a|_*=0$ implies $a=0$\,.)

 Using non-negativity of elements of ${\Jacobi}$,
 we have
\[ |({\Jacobi}(\vec{x}_c) \, \vec{z})_I|
\le\sum_{{J}\in{\cal I}_d}
{\Jacobi}(\vec{x}_c)_{{I}{J}}\,|z_J|,
\ \ \ I\in{\cal I}_d\,.
 \]
 Hence
\eqnb
\eqna{Jlinbd}
|{\Jacobi}(\vec{x}_c) \, \vec{z}|_*
\le \lambda |\vec{z}|_*\,,
\ \ \vec{{z}}\in\cmplx^{{\cal I}_d}.
\eqne

 We now see that \prpu{chrfcnconv}
 is a consequence of the following,
 which is proved in the proof of \cite[Proposition 4.4]{HHK93}.
\lemb
\lema{rec2conv}
 Let $p\in\nintegers$, and let $H^{(n)}:\ \cmplx^p\to\cmplx^p$,
 $n\in\nintegers$, be a sequence of polynomials with positive coefficients,
 and each term with degree $2$ or larger,
 which satisfies a recursion relation
\eqnb
\eqna{recH}
 H^{(n+1)}=H^{(1)}\circ H^{(n)},\ \ n\in\nintegers.\eqne

 Let $\partial H^{(1)}$ be the Jacobi matrix of $H^{(1)}$:
\[ \partial H^{(1)}_{ij}(z)=\frac{\partial H^{(1)}_i}{\partial z_j}(z),
\ \ i,j\in 1,\cdots,p,\ z\in\cmplx^p. \]
 Assume that $a\in\preals^p$ is a fixed point of $H^{(1)}$,
 and that
 there exists a norm $|\cdot|_*$ on $\cmplx^p$ and $\lambda>1$ such that
\eqnb
\eqna{Jlinbd0}
 |\partial H^{(1)}(a) z|_* \le \lambda |z|_*\,,
\ \ z\in\cmplx^p,
\eqne
 and that $x\in\preals^p$ is a point for which
 there exist positive constants $\rho$ and $C$, satisfying $\rho<1$,
 such that
\eqnb
\eqna{criticaltrajectory}
 |H^{(n)}_{i}(x)-a_i| \le C\rho^n,\ \ i=1,\cdots,p,\ n\in\pintegers,
\eqne
 and for which the limit
\eqnb
\eqna{scaledmeansteps0}
 \Lambda(x)=\limf{n} \lambda^{-n} \partial H^{(n)}(x)
\eqne
 exists.

 Then there exists an entire function
 $\dsp H^{*}=(H^{*}_1,\cdots,H^{*}_p):\ \cmplx^p\to\cmplx^p$
 such that
 \[ \limf{n}
 H^{(n)}(x_1\exp(\lambda^{-n}t_1),\cdots,x_p\exp(\lambda^{-n}t_p))
= H^{*}(t), \]
 uniformly in $t=(t_1,\cdots,t_p)$ on any compact set of $\cmplx^p$.

 $H^{*}$ satisfies
\[ H^{*}(\lambda t)=H^{(1)}(H^{*}(t)),\ \ t\in\cmplx^p, \]
 and
\[ \frac{\partial H^{*}_i}{\partial t_j}(0)=x_j \Lambda_{ij}(x),
\ \ i,j\in \{1,2,\cdots,p\}. \]
 Furthermore, if $a_{i}=0$ for some $i\in\{1,\cdots,p\}$, then
 $H^{*}_i(t)=0$\,.

\leme
\remb
 The proof of \cite[Proposition 4.4]{HHK93} is for $3$SG,
 but the proof is valid for the general case of \lemu{rec2conv}.
 In the proof of \cite[Proposition 4.4]{HHK93},
 \cite[Theorem 3.5]{HHK93} is referred to,
 which should be replaced,
 in the proof of \lemu{rec2conv},
 by $\dsp \limf{n} H^{(n)}(x)=a$, which is a consequence of
 the assumption \eqnu{criticaltrajectory}.
 Similarly, 
 Proposition 3.8, Proposition 4.3, and (4.4) in \cite{HHK93}
 are replaced by \eqnu{criticaltrajectory},
 \eqnu{scaledmeansteps0},
 and \eqnu{recH}, respectively.

\DDD\reme

 To apply \lemu{rec2conv} to the proof of \prpu{chrfcnconv},
 we only need to note that
 \prpu{RG}, \eqnu{Jlinbd}, \lemu{criticaltrajectory},
 and
 \prpu{scaledmeansteps},
 correspond to
 \eqnu{recH},
 \eqnu{Jlinbd0},
 \eqnu{criticaltrajectory},
 and
 \eqnu{scaledmeansteps0},
 respectively.

\QED\prfe



\prfofb{\protect\thmu{scaledlength}}
 Assume
 that $x_{c,I}\ne 0$ and $\vec{x}\in\Dom(\vec{x}_c)$\,.

 Denote by $\dsp p_{\vec{x},n,{I}}$,
 the joint distribution of
 $\dsp (\lambda^{-n} {s}_{{J}}$, ${J}\in{\cal I}_d)$
 under
 $\dsp {\mu}_{\vec{x},n,{I}}$\,.
 Its generating function is expressed, with
 the definitions \eqnu{genfcn} and \eqnu{mun}, as
\eqnb
\eqna{scaledensemblechrfcn}
 \int_0^{\infty} e^{\vec{{t}}\cdot \vec{{\xi}}}
 p_{\vec{x},n,{I}}[d\vec{{\xi}}]
= \frac{{X}_{n,{I}}(\vec{x}(\vec{{t}}))}{
{X}_{n,{I}}(\vec{x})}\,,
\ \ \vec{{t}}\in\cmplx^{{\cal I}_d},
\eqne
 where the notation $\dsp \vec{x}(\vec{{t}})$
 is as in \eqnu{xt}.
 Since the probability measure in question is a measure on a finite set,
 the integration over $\vec{{\xi}}$ in \eqnu{scaledensemblechrfcn}
 is actually a finite summation, hence the quantity is defined
 for all complex $\vec{{t}}$.

 Using \prpu{chrfcnconv} and (DA\ref{DA2})
 in \eqnu{scaledensemblechrfcn}, we see that the right hand side
 of \eqnu{scaledensemblechrfcn} converges to
 $\dsp {\varphi}_{{I}}^*(\vec{t})=
{\varphi}_{\vec{x},{I}}^*(\vec{t})$ of \prpu{chrfcnconv},
 uniformly in $\vec{{t}}$ on compact sets.
 Hence $\dsp p_{\vec{x},n,{I}}$
 converges weakly to a probability measure $\dsp p^*_{\vec{x},{I}}$
 whose generating function is
 $\dsp {\varphi}^*_{{I}}(\vec{{t}})$.

 We therefore see that
 the claims in \thmu{scaledlength},
 except those on the existence and positivity of the density
 of $\dsp\bar{p}^*_{\vec{x},(1)}$, are
 direct consequences of \prpu{scaledmeansteps} and
 \prpu{chrfcnconv}.
 That \eqnu{definingchrfcn} determines $\dsp\vec{\varphi}{}^*$ follows from
 the fact that its Taylor coefficients are determined recursively
 by \eqnu{definingchrfcn}.
 (Note that ${\Phi}_{{I}}$ is a polynomial with each term
 having degree $2$ or larger.)


\smallskip\par

 To prove that $\dsp \bar{p}^*_{\vec{x},I}$ has a $C^{\infty}$
 density, we prepare the following.

 Denote by $\dsp \bar{p}_{\vec{x},n,I}$,
 the distribution of $\lambda^{-n} L$
 under $\dsp {\mu}_{\vec{x},n,{I}}$\,.
 Its generating function is, from \eqnu{sL} and \eqnu{scaledensemblechrfcn},
\eqnb
\eqna{scaledLchrfcn}
 \int_0^{\infty} e^{t \xi}
 \bar{p}_{\vec{x},n,I}[d\xi]
= \frac{{X}_{n,I}(\vec{x}(t \vec{e}))}{
{X}_{n,I}(\vec{x})}\,,
\ \ t\in\cmplx,
\eqne
 where $\vec{e}=(e_J,\; J\in{\cal I}_d)$ is given by
\eqnb
\eqna{eJ}
e_J=|J|.
\eqne

 According to what has been proved below \eqnu{scaledensemblechrfcn},
 \eqnu{scaledLchrfcn} converges to an entire function
\eqnb
\eqna{gI}
 g_I(t)=g_{\vec{x},I}(t)={\varphi}_{I}^*(t\vec{e})
\eqne
 uniformly in $t$ on compact sets of $\reals$, and
 $\dsp \bar{p}_{\vec{x},n,{I}}$
 converges weakly to a probability measure $\dsp \bar{p}^*_{\vec{x},{I}}$
 whose generating function is $g_{I}$\,.

 \prpu{chrfcnconv} also implies that
\eqnb
\eqna{scaledmean}
 x_{c,I}\,\frac{d g_{I}}{d t}(0)
=x_{c,I}\,\int_0^{\infty} \xi \bar{p}^*_{\vec{x},I}[d\xi]
=\sum_{J\in{\cal I}_d} e_J \,x_{J}\, \Lambda_{IJ}(\vec{x}),
\ \ I\in {\cal I}_d\,,
 \eqne
 and
\eqnb
\eqna{limiteqn}
 x_{c,I} g_I(\lambda t)={\Phi}_{I}(\vec{x}_c\vec{g}(t)),
\ \ \ I\in{\cal I}_d,\ t\in\cmplx,
 \eqne
 where, the notation of the variable for $\Phi_I$
 is as in \thmu{scaledlength},
 and we also defined
\[ g_{J}=0,\ \ \mbox{ if } x_{c,J}=0\,. \]



\lemb
\lema{smoothdensity}
\itmb
\item
$\dsp M_I=M_I(\vec{x}):=\int_0^{\infty}\xi\bar{p}^*_{\vec{x},I}[d\xi] >0,
\ \ \ I\in{\cal K}_{\vec{x}_c}\,,
\ \vec{x}\in\Xi_d\cap\Dom(\vec{x}_c).$

\item
 If $\vec{x}\in\Xi_d\cap\Dom(\vec{x}_c)$ and
 $I\in{\cal K}_{\vec{x}_c}$ then $\dsp \bar{p}^*_{\vec{x},I}$
 is not concentrated on
 a single point (i.e., has non-zero variance). Namely,
\[ V_I=V_I(\vec{x}):=\int_0^{\infty}(\xi-M_I(\vec{x}))^2
\bar{p}^*_{\vec{x},I}[d\xi] >0,
\ \ \ I\in{\cal K}_{\vec{x}_c}\,,\ \vec{x}\in\Xi_d\cap\Dom(\vec{x}_c). \]

\item
 If $\vec{x}\in\Xi_d\cap\Dom(\vec{x}_c)$ then
 there exist positive constants $C_1$ and $C_2$ such that
\[ \max_{I\in {\cal K}_{\vec{x}_c}}
 |g_{I}(\II t)|
\le C_2 \exp(-C_1 |t|^{1/d_w}),\ \ t\in\reals, \]
 where $\dsp d_w$ is as in \eqnu{dw}.


\itme
\leme
\prfb
\itmb
\item
 Using \eqnu{scaledmean} we have
\[
\int_0^{\infty} \xi \bar{p}^*_{\vec{x},I}[d\xi]
\ge e_{(1)} \frac{x_{(1)}}{x_{c,I}}\, \Lambda_{I\,(1)}(\vec{x}) >0, \]
 where we used \eqnu{eJ} for $e_{(1)}>0$,
 $\vec{x}\in\Xi_d\cap \Dom(\vec{x}_c)$ for $x_{(1)}>0$,
 and \eqnu{Lambdapos} for $\Lambda_{I\,(1)}(\vec{x})>0$\,.

\item
 By definition (\prpu{RG} and \eqnu{genfcn}), we have
\[ {\Phi}_{{I}}(\vec{{x}})
=\sum_{m\in \pintegers^{{\cal I}_d}}\,
b_{{I},m}\, \prod_{{J}\in{\cal I}_d}
 {x}_{{J}}{}^{m_{{J}}},
\ \ \vec{{x}}\in\cmplx^{{\cal I}_d},
\ {I}\in{\cal I}_d\,,\]
where
\[ b_{I,m}=\sharp\{ w\in W_I^{(1)}\mid \vec{s}(w)=m\},
\ \ \ I\in{\cal I}_d\,,\ m\in\pintegers^{{\cal I}_d}. \]
 Then \eqnu{limiteqn} implies
\eqnb
\eqna{definingchrfcnreals}
 x_{c,{I}} g_I(\lambda t)
={\Phi}_{{I}}(\vec{x}_c\,\vec{g}(t))
=\sum_{m\in \pintegers^{{\cal I}_d}}\,
C_{{I},m} \prod_{{J}\in{\cal I}_d} g_J(t)^{m_J},
\ \ \ t\in\cmplx,
\eqne
 where we wrote
\[ C_{{I},m}
=b_{{I},m} \,\prod_{J\in{\cal I}_d} x_{c,J}^{m_{J}}\,.
\]

 Now assume that
 $\dsp\bar{p}_{\vec{x},I}$ is concentrated on a single
 point for some $I\in{\cal K}_{\vec{x}_c}$\,.
 Then its generating function is
\eqnb
\eqna{nulassump}
 g_I(t)=e^{M_I t}. \eqne
 Differentiating \eqnu{definingchrfcnreals} by $t$ up to twice,
 putting $t=0$,
 and using \eqnu{nulassump} on the left hand side,
 we obtain equations, which eventually lead to
\[
(0\le)\ \ \sum_m C_{{I},m} (\lambda M_{{I}} - \sum_{{J}} m_{{J}} M_{{J}})^2
= - \sum_m C_{{I},m}
 \sum_{{J}} m_{{J}} V_{{J}}\,.\]
 Since $C_{{I},m}$, $m_{{J}}$, and $V_{{J}}$
 are non-negative, it follows that
 $\dsp V_{{J}}=0$ for all $\dsp{J}\in{\cal I}_d$ such that
 there exists $m\in \pintegers^{{\cal I}_d}$ satisfying
 $m_J>0$ and $m_K=0$ if $K\not\in{\cal K}_{\vec{x}_c}$\,.
 In other words,
 $\dsp V_{{J}}=0$ for all $\dsp {J}\in{\cal I}_d$ such that
 $\dsp{\Phi}_{{I}}(\vec{{x}})$
 has a term which contains
 a positive power of ${x}_{{J}}$
 and composed of those ${x}_{{K}}$ with $K\in{\cal K}_{\vec{x}_c}$\,.
 With (FP\ref{FPirred}) we see that $J=(1)$ has such property for
 all $I\in{\cal K}_{\vec{x}_c}$.
 Hence $V_{(1)}=0$, and $\bar{p}_{\vec{x},(1)}$ is concentrated
 on a single point.

 The arguments after \eqnu{nulassump} can now be repeated
 with $I=(1)$. Using also (FP\ref{FPirred}) again,
 we eventually see that
 \eqnu{definingchrfcnreals} implies
\[ x_{c,(1)} \exp(\lambda\,M_{(1)}\,t)
=\sum_{m\in\pintegers^{{\cal K}_{\vec{x}_c}}} C_{(1),m}
\exp(\sum_{J\in{\cal K}_{\vec{x}_c}} M_{{J}}\,m_{{J}}\,t),
\ \  t\in\cmplx. \]
 But ${\Phi}_{(1)}$ is a polynomial, hence the right hand side
 is actually a finite summation with positive coefficients,
 so this can hold only if
\[ \lambda\,M_{(1)}=\sum_{J\in{\cal K}_{\vec{x}_c}} M_{J}\,m_{J}, \]
 for all $m$ such that $C_{(1),m}>0$\,.
 \prpu{RG} implies that $C_{(1),m}>0$ for $m$ such that;
\itmb
\item
 $m_{(1)}=2$ and $m_{J}=0$, ${J}\ne (1)$,
\item
 $m_{(1)}=d+1$ and $m_{J}=0$, ${J}\ne (1)$.
\itme
 Therefore
 $2M_{(1)}= (d+1) M_{(1)}$\,,
 hence $M_{(1)}=0$ for $d\ge 2$,
 which is a contradiction, because
 \prpu{xconepos} implies
 $(1)\in{\cal K}_{\vec{x}_c}$, which,
 with the preceding result, implies
 $\dsp M_{(1)}>0$\,.

 This proves that if $I\in{\cal K}_{\vec{x}_c}$ and
 $\vec{x}\in \Dom(\vec{x}_c)\cap \Xi_d$, then
 $\dsp \bar{p}_{\vec{x},I}$ is not concentrated on
 a single point.

\item
 The preceeding result implies, by a standard argument relating
 positivity of variances and the absolute values of characteristic
 functions, that
 there exists $a>0$ such that
\[ |g_{I}(\II t)|<1,\ \ -a<t<a,\ t\ne 0,\ I\in{\cal K}_{\vec{x}_c}\,. \]
 Since characteristic functions are continuous, this further implies
 that there exist $a'>0$ and $0<C<1$ such that
\eqnb
\eqna{seed}
 |g_{I}(\II t)|\le C,
\ \ \frac{a'}{\lambda}\le |t|\le a',\ t\in\reals,
\ I\in{\cal K}_{\vec{x}_c}\,. \eqne
 Also \eqnu{definingchrfcnreals} and (FP\ref{FP2}) and
 the fact that
 each term in ${\Phi}_{{I}}$ has degree $2$ or more (\prpu{RG})
 imply (together with the fact that absolute values of characteristic
 functions are no greater than $1$),
\[
\max_{I\in{\cal K}_{\vec{x}_c}} |g_{I}(\II \lambda t)|
\le
\max_{I\in{\cal K}_{\vec{x}_c}} |g_{I}(\II t)|^2,
\ \ t\in\reals.\]
 This and \eqnu{seed} imply
\[
 |g_{I}(\II t)|\le C_2 \exp(-C_1 |t|^{1/d_w}),
\ \ t\in\reals,\ I\in{\cal K}_{\vec{x}_c}\,,
\]
 for some positive constants $C_1$ and $C_2$\,.

\itme
 This completes a proof of \lemu{smoothdensity}.

\QED\prfe

 The last statement in \lemu{smoothdensity} implies the existence of
 $C^{\infty}$ density $\bar{\rho}^*_{\vec{x},(1)}$
 of $\bar{p}^*_{\vec{x},(1)}$\,.

\smallskip\par

 To prove the positivity of $\bar{\rho}^*_{\vec{x},(1)}$\,,
 substitute
 $\dsp g_{(1)}(t)
=\int_0^{\infty} e^{t\xi} \bar{\rho}^*_{\vec{x},(1)}(\xi)\,d\xi$
 in \eqnu{limiteqn}, to find
\[ \lambda^{-1}\bar{\rho}^*_{\vec{x},(1)}(\lambda^{-1} \xi)
=x_{c,(1)} (\bar{\rho}^*_{\vec{x},(1)}*\bar{\rho}^*_{\vec{x},(1)})(\xi)
+x_{c,(1)}{}^d
 (\bar{\rho}^*_{\vec{x},(1)}*\bar{\rho}^*_{\vec{x},(1)}*\cdots
*\bar{\rho}^*_{\vec{x},(1)})(\xi)
+\mbox{ non-negative terms},
\ \ \ \xi\ge 0\,, \]
 where the operation $*$ denotes convolution, and we also used
 \prpu{RG}.
 Note first that since $\bar{\rho}^*_{\vec{x},(1)}$ is continuous
 and  $M_{(1)}>0$, there is a $\xi_0>0$ for which
 $\bar{\rho}^*_{\vec{x},(1)}(\xi)>0$ in a neighborhood of $\xi_0$\,.
 Furthermore, the above relation implies that
 if $\bar{\rho}^*_{\vec{x},(1)}$ is positive in a neighborhood of
 $\xi_1,\xi_2\ge0$, then it also is in a neighborhood of
 $\lambda^{-1}(\xi_1+\xi_2)$.
 With \prpu{lambdaapriori} and
 the continuity of $\bar{\rho}^*_{\vec{x},(1)}$\,,
 we therefore see that
 $\bar{\rho}^*_{\vec{x},(1)}$ is positive in a neighborhood of $0$\,.
 The above relation also implies that
 if $\bar{\rho}^*_{\vec{x},(1)}$ is positive in a neighborhood of
 $\xi_1,\xi_2,\cdots,\xi_{d+1}\ge0$, then it also is in a neighborhood of
 $\lambda^{-1}(\xi_1+\xi_2+\cdots+\xi_{d+1})$.
 With \prpu{lambdaapriori}, we inductively see that
 $\bar{\rho}^*_{\vec{x},(1)}$ is positive on $(0,\infty)$.

 This completes a proof of \thmu{scaledlength}.

\QED\prfofe



%----------------------------
\subsection{Exponent for mean square displacement.}
%----------------------------

 Here we prove \thmu{number} and \thmu{msd}.
 Since the proofs are similar for the full model and the restricted model,
 we will concentrate on the full model.

 We introduce the following notation which we use
 throughout this section.

 Define $\nu:\ W^{(0)}\to \pintegers\cup\{\infty\}$ by
\eqnb
\eqna{scalerange}
 \nu(w)=\min\{n\in\pintegers\cup\{\infty\} \mid
 w(k)\in G_n\,,\ k=0,1,2,\cdots,L(w) \},\ \ w\in W^{(0)}.
\eqne
 Note the obvious relation
\eqnb
\eqna{Lnurel}
2^{\nu(w)-1}<L(w)\le d (d+1)^{\nu(w)},\ \ \ w\in W^{(0)}.
\eqne

 For $w\in W^{(0)},$ let $\norm{w}=\max\{|w(k)|\mid k=0,1,2,\cdots,L(w)\}$,
 where
 $|\cdot|$ denotes the (Euclidean) length in $\reals^d\supset F$.
 By definition,
\eqnb
\eqna{nueqnorm}
 2^{\nu(w)-1}<\norm{w}\le 2^{\nu(w)}.
\eqne



\subsubsection{Tauberian type estimates and number of paths.}
 Here we prove \thmu{number} for the full model.

 Let $\lambda$ be as before,
 $\beta_c$ be a critical point of the full model, and
 $\vec{x}_{can}(\beta_c)=\vec{x}_{can,{\cal I}_d}(\beta_c)$
 be as in \eqnu{beta}.
\prpb
\prpa{pathnumberdensityforPSAW}
 Define, for $b>0$, $n\in\pintegers$, $\xi\in\reals$,
 $\dsp c_n=b\lambda^{-n}\sqrt{n}$
 and $\dsp h_n(\xi)=\frac1{\sqrt{2\pi} c_n} e^{-\xi^2/(2c_n^2)}$.
 Then for sufficiently large $b$ it holds that
\[ \limf{n} (\bar{p}_{\vec{x}_{can}(\beta_c),n,I}*h_n)(\xi)
=\bar{\rho}^*_{\vec{x}_{can}(\beta_c),I}(\xi),
\ \ I\in{\cal K}_{\vec{x}_c}, \]
 uniformly in $\xi\in\reals$,
 where
\[ (\bar{p}_{\vec{x},n,I}*h_n)(\xi)
= \int_{\reals} h_n(\xi-\eta)\, \bar{p}_{\vec{x},n,I}(d\eta)
\]
 is a convolution.

\prpe
\prfb
 For $n\in\pintegers$ and  $I\in{\cal I}_d$\,,
 denote the characteristic function of
 $\dsp (\bar{p}_{\vec{x}_{can}(\beta_c),n,I}*h_n)(\xi)\,d\xi$
 by $\phi_{n,I}$\,.
 Then
\eqnb
\eqna{smeared}
 \phi_{n,I}(t)=\int e^{\II \xi t}
 (\bar{p}_{\vec{x}_{can}(\beta_c),n,I}*h_n)(\xi)\,d\xi
= \bar{\varphi}_{\vec{x}_{can}(\beta_c),n,I}(t) e^{-c_n^2t^2/2},
\eqne
 where, using \eqnu{beta}, \eqnu{ZX}, \eqnu{eJ}, \eqnu{xt}
 in \eqnu{scaledLchrfcn}, we find
\[ \bar{\varphi}_{\vec{x}_{can}(\beta_c),n,I}(t)=
\frac{Z_{n,I}(\beta_c-\II \lambda^{-n}t)}{Z_{n,I}(\beta_c)}\,,
\ \ t\in\cmplx. \]
 According to what is proved for \eqnu{gI}, we have
\eqnb
\eqna{limg}
 \limf{n} \bar{\varphi}_{\vec{x}_{can}(\beta_c),n,I}(t)
= g_{\vec{x}_{can}(\beta_c),I}(t),
 \eqne
 uniformly in $t$ on compact sets.
 Let
 $\dsp A=\{t\in\cmplx\mid \Im t\ge 0,\ \lambda^{-1}\le |t|\le 1\}$.
 Then \lemu{smoothdensity} and (CS\ref{CSfull}) implies
 $\dsp \sup_{t\in A} |g_{\vec{x}_{can}(\beta_c),I}(\II t)| <1$
 for $I\in {\cal K}_{\vec{x}_c}$\,.
 With (CS\ref{CSfull}) we therefore see that
 for sufficiently small positive  $\epsilon>0$
 there exists an integer $n_1$ such that
\eqnb
\eqna{Zscaling}
 |Z_{n,I}(\beta_c-\II \lambda^{-n}t)|
=Z_{n,I}(\beta_c) |g_{\vec{x}_{can}(\beta_c),n,I}(t)|
<x_{c,I}-\epsilon,\ \ \ I\in{\cal K}_{\vec{x}_c}\,,
\ n=n_1,n_1+1,\cdots,\ I\in{\cal K}_{\vec{x}_c}\,,\ t\in A.
 \eqne

 Define $\vec{x}'=(x'_I;\; I\in{\cal I}_d)$ by
\[ x'_I=\left\{ \arrb{ll} x_{c,I}-\epsilon,& I\in{\cal K}_{\vec{x}_c}\,,
\\ 0 & I\not\in{\cal K}_{\vec{x}_c}\,. \arre\right. \]
 Then $\vec{x}'\in D^o\setminus\{0\}$ by \thmu{D}.
 Since $D^o$ is an open set in $\Xi_d$\,, there exists
 $\delta>0$ such that
 if we define $\vec{x}''=(x''_I;\; I\in{\cal I}_d)$ by
\[ x''_I=\left\{ \arrb{ll} x_{c,I}-\epsilon,& I\in{\cal K}_{\vec{x}_c}\,,
\\ \delta & I\not\in{\cal K}_{\vec{x}_c}\,, \arre\right. \]
 then $\vec{x}''\in D^o$.
 Also, by the definition of ${\cal K}_{\vec{x}_c}$ and (CS\ref{CSfull}),
 there exists $n_0\ge n_1$ such that
\eqnb
\eqna{Zscaling2}
 |Z_{n,I}(\beta_c-\II \lambda^{-n}t)|\le Z_{n,I}(\beta_c)
<\delta,\ \ \ I\not\in{\cal K}_{\vec{x}_c}\,,
\ n=n_0,n_0+1,\cdots,\ t\in \reals.
 \eqne
 \prpu{lowtempphasefreeenergy}, \eqnu{Zscaling}, and \eqnu{Zscaling2}
 then imply, with $\vec{x}''\in D^o$, that there exists
 positive constants $C_1$ and $C_2$ such that
\eqnb
\eqna{Zscaledasymp}
\arrb{l}\dsp |Z_{m+n,I}(\beta_c-\II \lambda^{-n}t)|
=|X_{m,I}(\vec{Z}_n(\beta_c-\II \lambda^{-n}t)|
\le X_{m,I}(\vec{x}'')\le C_1 e^{-C_2 2^m},
\\ \dsp
\ \ \ m\in\pintegers,\ n=n_0,n_0+1,\cdots,\ I\in{\cal I}_d,\ t\in A.
\arre
\eqne

 Let $n$ be an integer satisfying $n> n_0$\,, and $t$ be a real
 satisfying $|t|\in[1,\lambda^{n-n_0-1})$,
 and $\dsp m=\left[\frac{\log|t|}{\log \lambda}\right]+1$, 
 where $[x]$ is the largest integer not exceeding $x$.
 Then $n-m\ge n_0$ and $\lambda^{-m} t\in A$\,.
 Hence \eqnu{Zscaledasymp} and \eqnu{smeared} imply
\[\arrb{l}\dsp | \phi_{n,I}(t) | \le
\frac{|Z_{m+n-m,I}(\beta_c-\II \lambda^{-(n-m)} (\lambda^{-m}t))|}{
Z_{n,I}(\beta_c)}
 \le \frac{C_1}{Z_{n,I}(\beta_c)}\, e^{-C_2 2^m}
 \le \frac{C_1}{Z_{n,I}(\beta_c)}\, e^{-C_2 |t|^{1/d_w}},
\\ \dsp \ \ n>n_0,\ I\in{\cal I}_d\,,\ |t|\in[1,\lambda^{n-n_0-1}).
\arre \]
 On the other hand, we have $\dsp\limf{n} Z_{n,I}(\beta_c)=x_{c,I}$\,,
 and $\dsp \limf{n} \phi_{n,I}(t)= g_{\vec{x}_{can}(\beta_c),I}(t)$,
 $t\in\reals$,
 by \eqnu{limg}.
 Hence the dominated convergence theorem implies
\[ \limf{n} \int_{\reals} |\chi_{[-\lambda^{n-n_0-1},\lambda^{n-n_0-1}]}(t)
\phi_{n,I}(t)-g_{\vec{x}_{can}(\beta_c),I}(t)|\,dt=0,
 I\in{\cal K}_{\vec{x}_c}\,, \]
 where $\chi_A$ denotes the indicator function of a set $A$.

 On the other hand,
\[\arrb{l} \dsp
 \int_{\reals} |\chi_{[-\lambda^{n-n_0-1},\lambda^{n-n_0-1}]}(t)
\phi_{n,I}(t)-\phi_{n,I}(t)|\,dt
\le 2 \int_{\lambda^{n-n_0-1}}^{\infty} e^{-c_n^2t^2/2}\,dt
\\ \dsp 
\le 2 \int_{\lambda^{n-n_0-1}}^{\infty} t e^{-c_n^2t^2/2}\,dt
= \frac{2}{b^2n} e^{-n (b^2 \lambda^{-2n_0-2}/2- 2\log\lambda)}
\ \to\ 0,\ \ n\to\infty,
\arre \]
 for sufficiently large $b$.
 Therefore
\[ \limf{n} \int_{\reals}
 |\phi_{n,I}(t)- g_{\vec{x}_{can}(\beta_c),I}(t)|\,dt =0\,. \]
 Using the inversion formula we therefore have
\[ \sup_{\xi\in\reals}
|(\bar{p}_{\vec{x}_{can}(\beta_c),n,I}*h_n)(\xi)
-\bar{\rho}^*_{\vec{x}_{can}(\beta_c),I}(\xi)|
\le \int_{\reals} e^{-\II \xi t}
 |\phi_{n,I}(t)- \bar{\varphi}_{\vec{x}_{can}(\beta_c),I}(t)|\,dt
\ \to\  0,\ \ n\to\infty. \]

\QED\prfe

 We will use the following in \secu{LandSpaths}.
\corb
\cora{pathnumberdensityforPSAW}
\itmb
\item
 There exists positive constants $C_1$ and $C_2$
 such that
\[ Z_{m+n,I}(\beta_c+ \lambda^{-n})
\le C_1 e^{-C_2 2^m},
\ \ \ m,n\in\pintegers,\ I\in{\cal I}_d\,. \]

\item
 There exists a positive constant $C$ such that
\[ Z_{n,I}(\beta_c-\lambda^{-n})\le C,
\ \ \ n\in\pintegers,\ I\in{\cal I}_d\,. \]

\itme
\core
\prfb
\itmb
\item
 Since $\vec{x}_{can}(\beta_c+ \lambda^{-n}t)\in D^o\cap \Xi_d$ for $t>0$,
 we may use \prpu{lowtempphasefreeenergy} in a similar way as in the proof
 of \eqnu{Zscaledasymp} to prove the first claim.

\item
 The second claim is a direct consequence of \prpu{chrfcnconv}
 for $\vec{t}=\vec{e}$ (defined in \eqnu{eJ}), combined
 with \eqnu{ZX}.

\itme
\QED\prfe

 \thmu{number} for the full model is an obvious
 consequence of the following.
\prpb
\prpa{number}
 If $\beta_c$ is the critical point, then
 there exist positive constants $C_i$\,, $i=1,2$, and real constants
 $C_i$, $i=3,4$, such that
\[ C_1 k^{C_3} e^{\beta_c k} \le N(k)\le C_2 k^{C_4} e^{\beta_c k},
\ \ k=1,2,3,\cdots. \]

\prpe
\prfb
 To prove the upper bound, define
\eqnb
\eqna{zetan}
 \zeta_n = \sum_{w\in W^{(0)};\; \nu(w)\le n} e^{-\beta_c L(w)},
\ \ n\in\pintegers.
\eqne
 Then a consideration similar to that in the proof of \prpu{RG}
 proves that there exists a polynomial $f_1:\ \reals^{{\cal I}_d}\to\reals$
 with positive coefficients, of maximal degree $d+1$,
 satisfying $f_1(\vec{0})=1$,
 such that
\eqnb
\eqna{zetanrec}
 \zeta_{n+1}\le f_1(\vec{Z}_n(\beta_c)) \zeta_n,\ \ n\in\pintegers.
\eqne
 The assumption (CS\ref{CSfull}) implies that
 $\vec{Z}_n(\beta_c)$ is bounded, hence there exist positive constants
 $A_1$ and $A_2\ge 1$ such that
\eqnb
\eqna{zetanbd}
 \zeta_n\le A_1 A_2^n\,,\ \  n\in\pintegers.
\eqne
 $L>2^{\nu-1}$ in \eqnu{Lnurel} therefore implies
\[ e^{-\beta_c k} N(k) \le
\sum_{w\in W^{(0)};\; L(w)\le k}e^{-\beta_c L(w)}
\le \zeta_{[\log_2 k]+1}\le A_1 A_2\, k^{\log_2 A_2},
\]
 which proves the upper bound.

 To prove the lower bound, let $b>0$ be a constant such that the
 bound in \prpu{pathnumberdensityforPSAW} holds, and
 let $\dsp d_n=\sqrt{2n\log\lambda} c_n
=\sqrt{2\log \lambda}\, b\, n\, \lambda^{-n}$, $n\in\pintegers$.
 Then \prpu{pathnumberdensityforPSAW} implies
\[ \arrb{l} \dsp
\limf{n} \sup_{\xi\in\reals}
 |\int_{[\xi-d_n,\xi+d_n]}
 h_n(\xi-\eta)\, \bar{p}_{\vec{x}_{can}(\beta_c),n,(1)}(d\eta)
-\bar{\rho}^*_{\vec{x}_{can}(\beta_c),(1)}(\xi)|
\\ \dsp
\le
 \limf{n} \sup_{\xi\in\reals} \int_{\reals\setminus[\xi-d_n,\xi+d_n]}
 h_n(\xi-\eta)\, \bar{p}_{\vec{x}_{can}(\beta_c),n,(1)}(d\eta)
\\ \dsp
\le \limf{n} h_n(d_n) =  \limf{n} \frac1{\sqrt{2\pi b^2 n }} =0\,.
\arre  \]
 With the positivity of $\bar{\rho}^*_{\vec{x}_{can}(\beta_c),(1)}$
 (\thmu{scaledlength}), this implies that there exists an integer
 $n_2$ and a positive constant $\epsilon$ such that
\[
\bar{p}_{\vec{x}_{can}(\beta_c),n,(1)}([\xi-d_n,\xi+d_n]) \ge c_n \epsilon,
\ \ n\ge n_2\,,\ \xi\in[\lambda^{-1},\lambda]. \]

 Let $k$ be a positive integer, and let $n$ be a positive integer
 satisfying $\lambda^{-n} k \in [1,\lambda)$.
 Assume that $k$ is sufficiently large so that $n\ge n_2$ and
 $d_n\le 1-\lambda^{-1}$.
 Then
\[ \bar{p}_{\vec{x}_{can}(\beta_c),n,(1)}
([\lambda^{-n} k -2d_n,\lambda^{-n} k ]) \ge c_n \epsilon. \]
 Note that we can construct an injection
 $\dsp \{w\in W^{(n)}_{(1)}\mid L(w)\le k\} \to
 \{w\in W^{(0)}\mid L(w)=k\}$ by extending the path.
 Hence
\[ Z_{n,(1)}(\beta_c) c_n \epsilon \le
 \sum_{w\in W_{(1)}^{(n)};\; k-2d_n\lambda^n\le L(w)\le k}
e^{-\beta_c L(w)} \le e^{2\beta_c d_n \lambda^n} e^{-\beta_c k} N(k). \]
 Since $\lambda^{-n} k \ge 1$ implies
 $\dsp n\le \frac{\log k}{\log \lambda}$\,,
 we have
\[ N(k) \ge e^{\beta_c k} k^{-1 -2b \beta_c \sqrt{2/ \log \lambda}}
 \sqrt{\log k} Z_{n,(1)}(\beta_c) \frac{b}{\sqrt{\log \lambda}}
 \epsilon, \]
 which implies the lower bound.

\QED\prfe


\subsubsection{Large deviation type estimates on
 long paths and short paths.}
\seca{LandSpaths}

 Define, for $n,m\in\pintegers$,
\[ U_{n,m}
=\sum_{\scriptstyle w\in W^{(0)}; \atop {\scriptstyle \nu(w)\le n,
 \atop \scriptstyle L(w)\ge\lambda^{n+(m/2)}}} e^{-\beta_c L(w)}
\ \ \ \ \ \ \ \ \mbox{ and }
\ \ \ \ \ \ \ \ V_{n,m}
=\sum_{\scriptstyle w\in W^{(0)}; \atop {\scriptstyle \nu(w)= n+1,
 \atop \scriptstyle L(w)\le\lambda^{n-m}}} e^{-\beta_c L(w)}. \]
\prpb
\prpa{LDestim}
 There exist positive constants $C'$ and $C''$ such that
\[ U_{n,m}\le C' A_2^n e^{-C''\lambda^{m/2}},\ \mbox{ and }
\ V_{n,m}\le C' A_2^n e^{-C'' 2^m},\ \ n,m\in\pintegers, \]
 where $A_2$ is as in \eqnu{zetanbd}.

\prpe
\prfb
 Let $C''$ be a positive constant satisfying
 $\dsp C''\le\frac{\lambda-\sqrt{\lambda}}{d+1}$\,.

 To prove the bound for $U_{n,m}$\,, define
\[ S_{n,m,I}=\sum_{w\in W^{(n)}_{I},\; L(w)\ge C'' \lambda^{n+(m/2)}}
 e^{-\beta_c L(w)},\ \ \ n,m\in\pintegers,\ I\in{\cal I}_d\,. \]
 Then, in a similar way as deriving \eqnu{zetanrec}, we find
\[ U_{n+1,m}\le f_1(\vec{Z}_n(\beta_c)) U_{n,m+1}
+ \sum_{I\in{\cal I}_d} S_{n,m,I}\,
 \frac{\partial f}{\partial x_I}(\vec{Z}_n(\beta_c))\, \zeta_n\,,
\ \ \ n,m\in\pintegers.\]
 This with \eqnu{zetanbd} implies that there exists $C_1>0$ such that
\[ A_2^{-n-1} U_{n+1,m}\le A_2^{-n} U_{n,m+1}
 + C_1 \sum_{I\in{\cal I}_d} S_{n,m,I}\,,
\ \ \ n,m\in\pintegers. \]

 Note that
 $\dsp L(w)\le d\,(d+1)^{\nu(w)}$
 in \eqnu{Lnurel} implies
\[ U_{n,m}=0,\ \ \lambda^{n+(m/2)}>d\,(d+1)^{n},\]
 which holds if
\[ m>\frac{2 \log d}{\log\lambda}+ 2 \alpha n, \]
 where we wrote $\dsp \alpha=\frac{\log (d+1)}{\log \lambda}-1$\,.
 (Note that \prpu{lambdaapriori} implies $\alpha>0$.)
 Therefore, if we write
\[ k_0(n,m)=\left[
 \frac{2\alpha n -m}{2\alpha+1}+\frac{2\log d}{(2\alpha+1)\log \lambda}
\right], \]
 then
\eqnb
\eqna{Uboundaux}
 U_{n,m} \le
 C_1 A_2^{n} \sum_{k=0}^{k_0(n,m)} \sum_{I\in{\cal I}_d} S_{n-k-1,m+k,I}\,,
\ \ n,m\in\pintegers.
 \eqne

 On the other hand,
\[ \arrb{l}\dsp
S_{n,m,I}\le \sum_{w\in W^{(n)}_{I},\; L(w)\ge C'' \lambda^{n+(m/2)}}
 e^{-(\beta_c-\lambda^{-n}) L(w)} e^{-C'' \lambda^{m/2}}
\le Z_{n,I}(\beta_c-\lambda^{-n}) e^{-C'' \lambda^{m/2}}
\\ \dsp
\le C e^{-C'' \lambda^{m/2}},
\ \ \ n,m\in\pintegers,\ I\in{\cal I}_d\,,
\arre  \]
 for a positive constant $C$, where we used \coru{pathnumberdensityforPSAW}.
 This and \eqnu{Uboundaux} imply
\[ U_{n,m}\le C C_1 \sharp {\cal I}_d \, A_2^n
 \sum_{k=0}^{\infty} e^{-C'' \lambda^{(m+k)/2}} .\]

 Note that there exists a constant $k_0$ such that
\[\lambda^{(m+k)/2}\ge \lambda^{(m-k_0)/2} + \lambda^{(k-k_0)/2},
\ \ m,k\in\pintegers. \]
 (This is because 
\[ \lambda^{(m+k)/2}- (\lambda^{m/2-k_0} + \lambda^{k/2-k_0})
=(\lambda^{m/2}-\lambda^{-k_0/2})(\lambda^{k/2}-\lambda^{-k_0/2})
 -\lambda^{-k_0} \]
 is increasing in $m$ and $k$, hence it is sufficient to choose $k_0$
 such that $1\ge 2\lambda^{-k_0/2}$.)
 Therefore there exists a positive constant $C'$ such that
\[ U_{n,m}\le C C_1 \sharp {\cal I}_d
 \sum_{k=0}^{\infty} e^{-C'' \lambda^{(k-k_0)/2}}
 \, A_2^n e^{-C'' \lambda^{(m-k_0)/2}}
 \le C' A_2^n e^{-C'' \lambda^{m/2}},\ \ n,m\in\pintegers. \]

\smallskip\par
 To prove the bound for $V_{n,m}$\,, define
\[ T_{n,m,I}=\sum_{w\in W^{(n)}_{I},\; L(w)\le \lambda^{n-m}}
 e^{-\beta_c L(w)},\ \ \ n,m\in\pintegers;\ m\le n,\ I\in{\cal I}_d\,, \]
 and write $\dsp \vec{T}_{n,m}=(T_{n,m,I}\,,\; I\in{\cal I}_d)$.
 Then, again in a similar way as deriving \eqnu{zetanrec}, we find
\[ V_{n,m}\le (f_1(\vec{T}_{n,m})-1)\zeta_n
=(f_1(\vec{T}_{n,m})-f_1(\vec{0}))\zeta_n\,. \]
 On the other hand, the condition $\dsp L(w)\le \lambda^{n-m}$ in
 the definition of $T_{n,m,I}$ implies
\[ T_{n,m,I}\le \sum_{w\in W^{(n)}_{I}} e^{-(\beta_c+\lambda^{m-n})L(w)+1}
=e Z_{n,I}(\beta_c+\lambda^{m-n}).\]
 With \coru{pathnumberdensityforPSAW} and \eqnu{zetanrec}
 we have the statement.

\QED\prfe

\prpb
\prpa{msd2}
For sufficiently large positive constant $\alpha$, it holds that
\[ \liminf_{k\to\infty} (\log k)^{s\alpha} k^{-s}
 E_k [\norm{w}^{s d_w}] >0,\ \ \ \ \ \mbox{and}
\ \ \ \ \  \limsup_{k\to\infty} (\log k)^{-s\alpha} k^{-s}
 E_k [\norm{w}^{s d_w}] <\infty,\]
 for all $s\ge 0$,
 where $E_k[\cdot]$ is the expectation defined in \thmu{msd}.

\prpe
\prfb
 For $k\in\nintegers$, define
 $\dsp \tilde{n}(k)=\left[\frac{\log k}{\log \lambda}\right]$.
 Obviously, $\dsp \lambda^{\tilde{n}(k)}\le k< \lambda^{\tilde{n}(k)+1}$.

 For positive integers $m$ and $k$ satisfying $m\le \tilde{n}(k)$,
 \prpu{LDestim} implies
\[ \arrb{l}\dsp
\sharp\{w\in W^{(0)} \mid L(w)=k,\ \nu(w)\le \tilde{n}(k)-m\}
\le e^{\beta_c k} U_{\tilde{n}(k)-m, 2m}
\\ \dsp
\le C' \exp(\beta_c k+(\tilde{n}(k)-m)\log A_2 -C'' \lambda^m)
\\ \dsp
\le C' \exp(\beta_c k+\log_{\log\lambda} A_2 \log k -C'' \lambda^m). 
\arre \]
 This and \prpu{number} imply that
 for sufficiently large $\alpha$ and for any real $\epsilon$,
 there exists $C>0$ such that
\[ \arrb{l} \dsp
 \tilde{P}_k[\nu(w)\le \tilde{n}(k)-\alpha \log\log k + \epsilon]
\le C' C_1^{-1} \exp((\log_{\log\lambda} A_2-C_3) \log k
-C'' \lambda^{-\epsilon} (\log k)^{\alpha\log\lambda})
\\ \dsp
 \le C e^{-(\log k)^3}, 
\ \ \ k\in\nintegers, \arre  \]
 where $\tilde{P}_k[\cdot]$ is as defined in \thmu{msd}.
 This, with \eqnu{nueqnorm}, implies for sufficiently large $\alpha$,
\[
 \limf{k} \tilde{P}_k[\norm{w}<(\log k)^{-\alpha/d_w} k^{1/d_w}]
\, e^{(\log k)^2} \le
\limf{k} \tilde{P}_k[2^{\nu(w)}<2(\log k)^{-\alpha/d_w} k^{1/d_w}]
\, e^{(\log k)^2}
 =0\,. \]
 Chebishev's inequality implies, for $s\ge 0$,
\[ E_k[\norm{w}^{s d_w}]\ge ((\log k)^{-\alpha} k)^s
(1-\tilde{P}_k[\norm{w}\le (\log k)^{-\alpha/d_w} k^{1/d_w}]). \]
 Therefore,
\[ \liminf_{k\to\infty} (\log k)^{s\alpha} k^{-s} E_k[\norm{w}^{s d_w}] >0,
\ \ s\ge0\,. \]

 Next,
 for positive integers $m$, $k$, and $\ell$, \prpu{LDestim} implies,
 with an obvious inequality
 $2^{m+\ell}\ge 2^{m-1} +2^{\ell-1}$, $m,\ell\in\pintegers$,
\[ \arrb{l}\dsp
\sharp\{w\in W^{(0)} \mid L(w)=k,\ \nu(w)= \tilde{n}(k)+m+\ell+2 \}
\le e^{\beta_c k} V_{\tilde{n}(k)+m+\ell+1, m+\ell}
\\ \dsp
\le C' \exp(\beta_c k+(\tilde{n}(k)+m+\ell+1)\log A_2 -C'' 2^{m+\ell})
\\ \dsp
\le C' \exp(\beta_c k+\log_{\log\lambda} A_2 \log k +m\log A_2-C'' 2^{m-1} )
 A_2^{\ell+1} \exp( -C'' 2^{\ell-1}).
\arre \]
 This and \prpu{number} imply that there exists a positive constant $C$
 such that
\[ \arrb{l}\dsp
\tilde{P}_k[\nu(w)\ge \tilde{n}(k)+\frac{\alpha}{\log 2}\log\log k]
 =\sum_{\ell=0}^{\infty}
 \tilde{P}_k[\nu(w)=\tilde{n}(k)+\ell+\frac{\alpha}{\log 2}\log\log k]
\\ \dsp
\le \frac{C'}{C_1A_2} \sum_{\ell=0}^{\infty} A_2^{\ell} e^{-C''2^{\ell-1}}
\exp((\log_{\log\lambda} A_2 -C_3)\log k
 +\alpha \log_2 A_2 \log\log k - \frac{C''}{8} (\log k)^{\alpha})
\\ \dsp
\le C \exp((\log_{\log\lambda} A_2 -C_3)\log k
 +\alpha \log_2 A_2 \log\log k - \frac{C''}{8} (\log k)^{\alpha}),
\ \ k\in\nintegers.
\arre \]
 This, with \eqnu{nueqnorm}, implies for sufficiently large $\alpha$,
\[
 \limf{k} \tilde{P}_k[\norm{w}\ge (\log k)^{\alpha/d_w} k^{1/d_w}]
\, e^{(\log k)^2} \le
\limf{k} \tilde{P}_k[2^{\nu(w)}\ge (\log k)^{\alpha/d_w} k^{1/d_w}]
\, e^{(\log k)^2}
 =0\,. \]

 Let $s\ge 0$.  Note that $\norm{w}\le L(w)$, which implies
\[ E_k[\norm{w}^{s d_w}]\le ((\log k)^{\alpha} k)^s
+ k^{s d_w} \tilde{P}_k[\norm{w}\ge (\log k)^{\alpha} k]. \]
 Therefore,
\[ \limsup_{k\to\infty} (\log k)^{-s\alpha} k^{-s} E_k[\norm{w}^{s d_w}]
 <\infty,\]

\QED\prfe


\subsubsection{Reflection principle.}

 The definition of pre-$d$SG in \eqnu{finitepreSG} and \eqnu{preSG}
 induces a natural coordinate system on $G$ which is an onto map
 $\pi:\ \{0,1,2,\cdots,d\}^{\pintegers}\to G$ defined as follows.
 For each $v_{0,i}=v_i\in G_0$ ($i=0,1,2,\cdots,d$)
 we assign a coordinate $(i,0,0,0,\cdots)$;
\[ \pi(i,0,0,0,\cdots)=v_{0,i},\ i=0,1,2,3,\cdots,d. \]

 For $n=0,1,2,\cdots$ and $i=0,1,2,3,\cdots,d$,
 put $G_{n,i}=G_n+2^n v_{0,i}$\,, and
 define a $1:1$ onto map $\iota_{n,i}:\ G_{n,i}\to G_n$ by
 $\iota_{n,i}(v)=v-2^n v_{0,i}$\,.
 $\iota_{n,i}$ naturally induces a $1:1$ onto map $B_{n,i}\to B_{n}$\,,
 which we also denote by $\iota_{n,i}$.

 We proceed with by induction in $n$ and assume that
 a coordinate system $\pi$ on $G_{n-1}$ ($=G_{n-1,0}$)
 has been defined for an $n\ge1$,
 in such a way that $\pi(v)\in G_{n-1}$ holds
 for any $v=(i_0,i_1,i_2,\cdots,i_{n-1},0,0,0,\cdots)$
 with $i_k\in\{0,1,2,\cdots,d\}$, $k=0,1,2,\cdots,n-1$.
 For $v\in G_{n-1,j}$, with $j\in\{0,1,2,\cdots,d\}$, define
\[ \pi(i_0,i_1,i_2,\cdots,i_{n-1},j,0,0,\cdots)=v,
\ \mbox{ if }
\ \pi(i_0,i_1,i_2,\cdots,i_{n-1},0,0,0,\cdots)=\iota_{n-1,j}(v). \]
 Note that this definition is compatible with
 $G_{n-1,0}=G_{n-1}\subset G_n$\,.

 Each point in $G\setminus\{O\}$ has exactly two coordinate
 representations, because
\[ \pi(j,j,\cdots,j,i,0,\cdots)=\pi(i,i,\cdots,i,j,0,\cdots)
\in G_{m,i}\cap G_{m,j}\,,\ \ 0\le i<j\le d,\ m\in\pintegers. \]
 Note also that if $\pi(i_0,i_1,i_2,\cdots)\in G_n$ then
 $i_k=0$, $k=n+1,n+2,\cdots$.

\smallskip\par

 We now define a reflection map with which we formulate a reflection principle
 (\thmu{RP}).
 For each $i=1,2,3,\cdots,d$ define
 $R_{0,i}:
\ \{0,1,2,\cdots,d\}\to \{0,1,2,\cdots,d\}$
 by
\[ \left\{\arrb{l}
R_{0,i}(0)=i, \\ R_{0,i}(i)=0, \\ R_{0,i}(j)=j,\ j\ne 0,i.
\arre\right. \]

 For $n=1,2,3,\cdots$ and $i=1,2,\cdots,d$, we define
 $1:1$ maps $R_{n,i}:\ G\to G$ and $\tilde{R}_{n,i}:\ G\to G$
 (`partial reflections' with respect to a hyperplane containing
 $\pi(j,j,\cdots,j,i,0,0,\cdots)\in G_{n-1,i}$\,, $j\ne 0,i$,
 and `perpendicular to $i$-th axis'), by:
\eqnb
\eqna{R}
 \arrb{l} \dsp R_{n,i}(\pi(x_0,x_1,x_2, \cdots)) \\
\dsp =\left\{\arrb{l}
\dsp \pi(x_0,x_1,x_2, \cdots),
\mbox{ if }
 \pi(x_0,x_1,x_2, \cdots,x_n,\cdots)\not\in G_{n-1,i},
\\ \dsp
 \pi(R_{0,i}(x_0),R_{0,i}(x_1),\cdots,R_{0,i}(x_{n-1}),x_n,0,0,0,\cdots),
 \\ \dsp \phantom{\pi(x_0,x_1,x_2, \cdots),}
\mbox{ if }
 \pi(x_0,x_1,x_2, \cdots,x_n,\cdots)\in G_{n-1,i}\ \mbox{ and }\ x_n=i,
\arre\right.\arre
 \eqne
 and
\eqnb
\eqna{tildeR}
 \arrb{l}\dsp \tilde{R}_{n,i}(\pi(x_0,x_1,x_2, \cdots)) \\
\dsp =\left\{\arrb{l}
\dsp \pi(x_0,x_1,x_2, \cdots),
\mbox{ if }
 \pi(x_0,x_1,x_2, \cdots,x_n,\cdots)
\not\in G_{n-1}\cup \iota_{n,i}^{-1}(G_{n-1}),
\\ \dsp \pi(R_{0,i}(x_0),R_{0,i}(x_1),\cdots,
R_{0,i}(x_{n-1}),x_n,R_{0,i}(x_{n+1}),0,0,\cdots),
 \\ \dsp \phantom{\pi(x_0,x_1,x_2, \cdots),\ }
\mbox{ if }
 \pi(x_0,x_1,x_2, \cdots,x_n,\cdots)\in G_{n-1}\cup \iota_{n,i}^{-1}(G_{n-1})
\ \mbox{ and } x_n=0\,.
\arre\right.\arre
 \eqne
 Note that $G_{n-1}$\,, $G_{n-1,i}$\,, and $\iota_{n,i}^{-1}(G_{n-1})$ are
 three copies of $G_{n-1}$ aligned in `$i$-th axis' direction, such that
\eqnb\eqna{A} G_{n-1}\cap G_{n-1,i}=\{
\pi(\shortstack{$0,1,2$\\ $0,0,0$},\cdots,0,\shortstack{$n$\\$i$},0,0,\cdots)
\}=\{
\pi(\shortstack{$0,1,2$\\ $i,i,i$},\cdots,
\shortstack{$n-1$\\$i$},0,0,0,\cdots)
\}, \eqne
 and
\eqnb\eqna{B} G_{n-1,i}\cap \iota_{n,i}^{-1}(G_{n-1})=\{
\pi(\shortstack{$0,1,2$\\ $0,0,0$},\cdots,0,\shortstack{$n+1$\\$i$},0,0,\cdots)
\}=\{
\pi(\shortstack{$0,1,2$\\ $i,i,i$},\cdots,i,\shortstack{$n$\\$i$},0,0,\cdots)
\}. \eqne
 Note also that, by construction all the points in $G_{n-1,i}$ can be
 written as $\pi(x_0,x_1,\cdots,x_n,0,0,\cdots)$ with $x_n=i$,
 those in $G_{n-1}$ as $\pi(x_0,\cdots,x_n,0,0,\cdots)$ with $x_n=0$,
 and those in $\iota_{n,i}^{-1}(G_{n-1})$ as
 $\pi(x_0,\cdots,x_n,x_{n+1},0,\cdots)$ with $x_n=0$ and $x_{n+1}=i$.

\prpb\prpa{Rproperties}
 $R_{n,i}$ and $\tilde{R}_{n,i}$ are $1:1$ maps.
 Moreover, the following hold.
\itmb
\item
 If $x\in G_{n-1}$ then $\dsp \tilde{R}_{n,i}(x)\in\iota_{n,i}^{-1}(G_{n-1})$.
\item
 If $\dsp (x,y)\in B_{n-1}$,
 then $\dsp (\tilde{R}_{n,i}(x),\tilde{R}_{n,i}(y))\in B$.
\item
 If $\dsp x\in G_{n-1}\cap G_{n-1,i}$
 then $\dsp R_{n,i}(x)=\tilde{R}_{n,i}(x)$.
\item
 If $x\in G_n$ then $\dsp R_{n,i}(x)\in G_n$\,.
\item
 If $\dsp x\in G_{n-1,i}\cap \bigcup_{j\ne 0,i} G_{n-1,j}$
 then $\dsp R_{n,i}(x)=x$.
\item
 If $\dsp (x,y)\in B_{n-1,i}$
 then $(R_{n,i}(x),R_{n,i}(y))\in B$.
\itme
\prpe
\prfb
 By definition, $R_{n,i}^2(x)=\tilde{R}_{n,i}^2(x)=x$, $x\in G$, hence,
 in particular,  $R_{n,i}$ and $\tilde{R}_{n,i}$ are $1:1$ maps.
\itmb
\item
 If $x\in G_{n-1}$ then $x$ has a coordinate of the form
 $x=\pi(x_0,x_1,\cdots,x_{n-1},0,0,0,\cdots)$,
 which, with \eqnu{tildeR}, implies
 $\dsp \tilde{R}_{n,i}(x)=\pi(x_0,x_1,\cdots,x_{n-1},0,i,0,\cdots)$, which
 is in $\dsp \iota_{n,i}^{-1}(G_{n-1})$\,.

\item
let $\dsp (x,y)\in B_{n-1}$\,.
 Then $x,y\in G_{n-1}$, hence their coordinates can be written as
\[ x=\pi(x_0,x_1,\cdots,x_{n-1},0,0,0,\cdots),
\ \ y=\pi(y_0,y_1,\cdots,y_{n-1},0,0,0,\cdots). \]
 With \eqnu{tildeR}, we have
 \[\tilde{R}_{n,i}(x)=
\pi(R_{0,i}(x_0),R_{0,i}(x_1),\cdots,R_{0,i}(x_{n-1}),0,i,0,0,\cdots)
\in \iota_{n,i}^{-1}(G_{n-1}), \]
 and a similar expression holds for $y$.
 Noting that $\iota_{n,i}^{-1}(G_{n-1})$ is a copy of $G_{n-1}$,
 it then suffices to prove
\[
(\pi(R_{0,i}(x_0),\cdots,R_{0,i}(x_{n-1}),0,0,\cdots),
\pi(R_{0,i}(y_0),\cdots,R_{0,i}(y_{n-1}),0,0,\cdots))
 \in B_{n-1}\,. \]
 Noting the definition of $B_{n-1}$, we may assume, from $(x,y)\in B_{n-1}$,
 that $x_{n-1}=y_{n-1}$ and 
\[ (\pi(x_0,x_1,\cdots,x_{n-2},0,0,0,\cdots),
\pi(y_0,y_1,\cdots,y_{n-2},0,0,0,\cdots))\in B_{n-2}\,. \]
 Inductively, it eventually turns out that it suffices to prove
\[(\pi(R_{0,i}(x),0,0,\cdots),\pi(R_{0,i}(y),0,0,\cdots))\in B_{0}
\ \ \ \mbox{if}\ \ (\pi(x,0,0,\cdots),\pi(y,0,0,\cdots))\in B_{0}\,, \]
 which is obvious from the definition of $F_0=(G_0,B_0)$.

\item
 Applying \eqnu{A} in \eqnu{R} and \eqnu{tildeR}, we have
\[ R_{n,i}(x)=R_{n,i}(
\pi(\shortstack{$0,1,2$\\ $0,0,0$},\cdots,0,\shortstack{$n$\\$i$},0,0,\cdots)
)=
\pi(\shortstack{$0,1,2$\\ $i,i,i$},\cdots,i,\shortstack{$n$\\$i$},0,0,\cdots),
\]
 and
\[ \arrb{l} \dsp \tilde{R}_{n,i}(x)=\tilde{R}_{n,i}(
\pi(\shortstack{$0,1,2$\\ $i,i,i$},\cdots,
\shortstack{$n-1$\\$i$},0,0,0,\cdots)
)=
\pi(\shortstack{$0,1,2$\\$0,0,0$},\cdots,0,0,\shortstack{$n+1$\\$i$},0,\cdots)
\\ \dsp
=
\pi(\shortstack{$0,1,2$\\ $i,i,i$},\cdots,i,\shortstack{$n$\\$i$},0,0,\cdots).
\arre
\]
 Hence $R_{n,i}(x)=\tilde{R}_{n,i}(x)$.

\item
 If $x\in G_n$ then $x$ has a coordinate
 $x=\pi(x_0,x_1,\cdots,x_n,0,0,\cdots)$.
 Then \eqnu{R} implies that $R_{n,i}(x)$ has a similar coordinate,
 hence $R_{n,i}(x)\in G_n$\,.

\item
 Noting \eqnu{A} and \eqnu{B}, we see that $x$ has a coordinate
 $x=\pi(j,j,\cdots,j,\shortstack{$n$\\$i$},0,0,\cdots)$, $j\ne 0,i$.
 Then \eqnu{R} implies $R_{n,i}(x)=x$.

\item
 This can be proved  in a similar way as (ii). 

\itme
\QED\prfe

 For each self-avoiding path $w\in W^{(0)}$
 satisfying $\dsp |w(L(w))| < 2^{\nu(w)-1}$,
 we want to assign a self-avoiding path $R(w)\in W^{(0)}$
 (`reflected path') such that
 $\dsp  |R(w)(L(R(w)))| > 2^{\nu(R(w))-1}$
 (the right hand side stands for the Euclidean distance of
 the endpoints of $R(w)$).
 This is possible using \eqnu{R} and \eqnu{tildeR}, as follows.

 Given  $w=(w(0),w(1), \cdots, w(L(w)))\in W^{(0)}$
 with the property
\eqnb\eqna{long}
w(L(w))\in G_{n-1}\setminus \bigcup_{i=1}^d G_{n-1,i} \,,
\ \mbox{ where }\ n=\nu(w),\eqne
 we shall define a path $R(w)$ as follows.

 Let $T(w)$ be a positive integer satisfying
\[ w(k)\in G_{n-1},\ T(w)\le k\le L(w),\ \mbox{ and }
\ w(T(w)-1)\not\in G_{n-1}\,.\]
 The condition \eqnu{long} implies that such an integer (uniquely) exists.
 Since $\dsp w(T(w))\in G_{n-1}$,
 $\dsp w(T(w)-1)\in \bigcup_{i=1}^d G_{n-1,i}$\,.
 Let $i(w)\in\{1,2,\cdots,d\}$ be such that
 $\dsp w(T(w)-1)\in G_{n-1,i(w)}$\,.
 Clearly, such an integer is also unique.
 Also the definitions of $T(w)$ and $i(w)$ imply
\eqnb\eqna{wT} w(T(w))\in G_{n-1}\cap G_{n-1,i(w)}\,.\eqne
 Define $R(w)$ by
\[ R(w)(k)=\left\{\arrb{ll}
   \dsp R_{n,i(w)}(w(k)), & 0\le k<T(w),
\\ \dsp \tilde{R}_{n,i(w)}(w(k)), & T(w)\le k\le L(w).
\arre\right. \]

\thmb
\thma{RP}
 For each $w\in W^{(0)}$
 satisfying $\dsp |w(L(w))| < 2^{\nu(w)-1}$,
 $R(w)\in W^{(0)}$ (i.e., is a self-avoiding path starting from $O$)
 which satisfies $L(R(w))=L(w)$,
 $\nu(R(w))=\nu(w)+1$, and $2^{\nu(w)}< |R(w)(L(w))|$\,.

\thme
\prfb
 The non-trivial part of the claim is that $R(w)$
 is a self-avoiding path.
 All the other properties are obvious from the definition of the reflections.

 We first prove that $R(w)$ is self-avoiding, when $w$ is
 self-avoiding.
 Let $n=\nu(w)$.
 Note that \eqnu{wT} and \prpu{Rproperties} (iii) imply
\eqnb\eqna{RwTw} R_{n,i(w)}(w(T(w)))
=\tilde{R}_{n,i(w)}(w(T(w))). \eqne
 Since $R_{n,i}$ and $\tilde{R}_{n,i}$ are $1:1$ maps
 (\prpu{Rproperties}), the path segments
 $\dsp w_1=\{ R(w)(k),\ 0\le k\le T(w)\}$
 and
 $\dsp w_2=\{ R(w)(k),\ T(w)\le k\le L(w)\}$
 are self-avoiding.

 The definition of $T(w)$ implies
 $w(k)\in G_{n-1}$, $T(w)\le k \le L(w)$,
 which, with \prpu{Rproperties} (i), implies
 $\dsp \tilde{R}_{n,i}(w(k))\in\iota_{n,i}^{-1}(G_{n-1})$,
 hence, in particular,
 $\dsp w_2\cap G_n=\{R_{n,i}(w(T(w)))\}$.
 On the other hand, \prpu{Rproperties} (iv) implies $w_1\subset G_n$\,.
 Therefore $w_1$ and $w_2$ are mutually avoiding.
 This proves that $R(w)$ is self-avoiding.

 We are left with proving that $R(w)$ is a path, i.e.,
\eqnb\eqna{pathclaim}
 (R(w)(k),R(w)(k+1))\in B,\ k=0,1,2,\cdots,L(w)-1\,. \eqne
 Definition of $T(w)$ and \prpu{Rproperties} (ii)
 imply \eqnu{pathclaim} for $T(w)\le k< L(w)$.
 Hence we may assume $0\le k<T(w)$.
 Definition of $T(w)$ and \eqnu{RwTw} imply
 $R(w)(k)=R_{n,i(w)}(w(k))$, $0\le k\le T(w)$.
 If $\dsp w(k)\not\in G_{n-1,i(w)}$\,, then
 $(w(k),w(k+1))\in B$ and $n=\nu(w)$ and \eqnu{wT} 
 with self-avoiding property of $w$ imply
 $\dsp w(k+1)\not\in G_{n-1,i(w)}$ or
 $\dsp w(k+1)\in G_{n-1,i(w)}\cap
 \bigcup_{j\ne 0,i(w)} G_{n-1,j}$\,.
 The definition of $R_{n,i(w)}$ and \prpu{Rproperties}(v) imply
 that in either case $\dsp R_{n,i(w)}(w(k+1))=w(k+1)$.
 Also $\dsp w(k)\not\in G_{n-1,i(w)}$ implies
 $\dsp R_{n,i(w)}(w(k))=w(k)$.
 Hence in this case \eqnu{pathclaim} holds.

 The remaining case is when $0\le k<T(w)$ and
 $\dsp w(k)\in G_{n-1,i(w)}$ hold.
 If $\dsp w(k+1)\not\in G_{n-1,i(w)}$, then a similar reasoning
 as above applies. Therefore we may also assume
 $\dsp w(k+1)\in G_{n-1,i(w)}$\,.
 Hence $(w(k),w(k+1))\in B_{n-1,i(w)}$, in which case
 \prpu{Rproperties} (vi) applies and we have \eqnu{pathclaim}.

\QED\prfe
\corb
\cora{RP}
 If $s\ge 0$ and $k\in\nintegers$,
\[ E_k[2^{s (\nu(w)-1)};\ |w(k)|\le 2^{\nu(w)-1}]
\le E_k[2^{s (\nu(w)-1)};\ |w(k)|\ge 2^{\nu(w)-1}] , \]
 where $E_k[\cdot]$ is the expectation defined in \thmu{msd}.

\core
\prfb
 Recalling the definition of the measure $\tilde{P}_k$\,,
 we find from \thmu{RP},
\[\arrb{l}\dsp
 E_k[2^{s (\nu(w)-1)};\ |w(k)|< 2^{\nu(w)-1}]
=\sum_{n=0}^{\infty} 2^{s(n-1)} \tilde{P}_k[ \nu(w)=n,\ |w(k)|< 2^{n-1}]
\\ \dsp
\le \sum_{n=0}^{\infty} 2^{s(n-1)} \tilde{P}_k[ \nu(w)=n+1,\ |w(k)|> 2^{n}]
= 2^{-s} E_k[2^{s(\nu(w)-1)};\ |w(k)|> 2^{\nu(w)-1}]
\\ \dsp
\le E_k[2^{s(\nu(w)-1)};\ |w(k)|> 2^{\nu(w)-1}].
\arre \]
 Hence
\[ \arrb{l}\dsp
E_k[2^{s (\nu(w)-1)};\ |w(k)|\le 2^{\nu(w)-1}]
= E_k[2^{s (\nu(w)-1)};\ |w(k)|= 2^{\nu(w)-1}]
+ E_k[2^{s (\nu(w)-1)};\ |w(k)|< 2^{\nu(w)-1}]
\\ \dsp
\le E_k[2^{s (\nu(w)-1)};\ |w(k)|\ge 2^{\nu(w)-1}] .
\arre \]

\QED\prfe

\smallskip\par

 We are now ready to prove \thmu{msd}.
\prfofb{\protect\thmu{msd}}
 Note an obvious bound
\[ |w(L(w))|\le \norm{w} \le 2^{\nu(w)},\ \ w\in W^{(0)}. \]
 This and \coru{RP} imply
\[ \arrb{l}\dsp
 2^{-s-1} E_k[\norm{w}^s]
\\ \dsp
\le 2^{-s-1} E_k[2^{s \nu(w)}]
= \frac12 (E_k[2^{s (\nu(w)-1)};\ |w(k)|\le 2^{\nu(w)-1}]
+ E_k[2^{s (\nu(w)-1)};\ |w(k)|> 2^{\nu(w)-1}])
\\ \dsp
\le E_k[2^{s (\nu(w)-1)};\ |w(k)|\ge 2^{\nu(w)-1}]
\le E_k[|w(k)|^s;\ |w(k)|\ge 2^{\nu(w)-1}]
\\ \dsp
\le E_k[|w(k)|^s]
\le E_k[\norm{w}^s],
\ \  \ s\ge 0,\ k\in\integers.
 \arre \]
 This and \prpu{msd2} further imply, for sufficiently large $\alpha$,
\[ \liminf_{k\to\infty} (\log k)^{s\alpha} k^{-s}
 E_k[|w(k)|^{s\, d_w}] >0,\ \ \ \ \ \mbox{and}
\ \ \ \ \  \limsup_{k\to\infty} (\log k)^{-s\alpha} k^{-s}
 E_k[|w(k)|^{s\, d_w}]<\infty.\]
 Namely, there exist constants $k_0$, $C$, and $C'$ such that
\[ s\log k-s\alpha\log\log k+C
 \le \log E_k[|w(k)|^{s\, d_w}]
 \le s\log k +s\alpha\log\log k +C',\ \ k\ge k_0,\ s\ge 0.\]
 This implies
\[ \limf{k} \frac1{\log k} \log E_k[|w(k)|^{s\, d_w}]=s,
\ \ s\ge 0, \]
 which completes a proof of \thmu{msd}.
\QED\prfofe




\section{Restricted model on $4$-dimensional {\sg}.}
\seca{deqfour}

 In this section, we consider the case $d=4$.
 The RG map $\dsp\vec{\Phi}$ (\prpu{RG}) is a map on $6$ dimensional space
 $\cmplx^{{\cal I}_4}$ where, as in \secu{definitions},
 ${\cal I}_4=\{(1),(1,1),(2),(3),(4),(1,2)\}$.
 (For convenience, we assign the second coordinate to $(1,1)$
 in this section.)

 We will consider the restricted self-avoiding walks,
 the self-avoding paths $w$ starting from $O$ with the property
 $s_J(w)=0$, $J\not\in {\cal K}_{res}$\,,
 where ${\cal K}_{res}=\{(1),(11) \}$ (see \eqnu{Kres} and \eqnu{WK}).
 We regard $\preals^{{\cal K}_{res}}=\{(x_{(1)},x_{(11)},0,\cdots,0)\mid
 x_{(1)},x_{(11)}\in\preals \}\subset \preals^{{\cal I}_4}$.

 To apply the results of previous sections for the restricted model,
 we use the following explicit properties of $\vec{\Phi}$.
\prpb
\prpa{Phi4}
 The map $\vec{\Phi}$ satisfies the following.
\itmb
\item
\label{Phi41}

 $\vec{\Phi}$ is a $6$ dimensional vector valued function whose components
 are polynomials in $6$ variables $(1)$, $(11)$, $(2)$, $(3)$, $(4)$, $(12)$
 with positive integer coefficients.
 The degree of each term in the polynomials are no less than $2$
 and no greater than $5$.

\item
\eqnb
\eqna{Phi4}
\arrb{rcl}
\dsp \Phi_{(1)}(x,y,0,0,0,0) &=& \dsp
 x^2 + 3x^3 + 6x^4 +6x^5 +12x^3 y +30 x^4 y +18 x^2 y^2
\\ \dsp  & & \dsp {}
 + 78x^3 y^2 +96x^2 y^3 +132 x y^4 +132 y^5,
\\ \dsp
\Phi_{(1,1)}(x,y,0,0,0,0) &=& \dsp
 x^4 +2x^5 +4x^3 y +13x^4 y +32 x^3 y^2 +88 x^2 y^3
\\ & & \dsp{}
 + 22y^4 +220 x y^4 +186 y^5.
\arre
\eqne

\item

 There exist polynomials $\Phi_{I,i}$\,, $I=(1),(11)$, $i=0,1,2,3,4$,
 of positive coefficients, such that
 $\Phi_{(1),0}$ contains a term $x_{(1)}^2$,
 $\Phi_{(11),0}$ contains a term $x_{(1)}^4$,
 and
\eqnb
\eqna{Phi40}
 \arrb{rcl} \dsp
\Phi_{(1)}(\vec{x})&=&
 \Phi_{(1),1}(\vec{x})\, x_{(1)}
+ \frac12 \Phi_{(1),2}(\vec{x})\, x_{(2)}
+ \frac13 \Phi_{(1),3}(\vec{x})\, x_{(3)}
+ \Phi_{(1),4}(\vec{x})\, x_{(11)}
+ \Phi_{(1),0}(\vec{x}),
\\ \dsp
\Phi_{(2)}(\vec{x})&=&
 \Phi_{(1),1}(\vec{x})\, x_{(2)}
+ \Phi_{(1),2}(\vec{x})\, x_{(3)}
+ \Phi_{(1),3}(\vec{x})\, x_{(4)}
+ \Phi_{(1),4}(\vec{x})\, x_{(12)}\,,
\\ \dsp
\Phi_{(11)}(\vec{x})&=&
 \Phi_{(11),1}(\vec{x})\, x_{(1)}
+ \frac12 \Phi_{(11),2}(\vec{x})\, x_{(2)}
+ \frac13 \Phi_{(11),3}(\vec{x})\, x_{(3)}
+ \Phi_{(11),4}(\vec{x})\, x_{(11)}
+ \Phi_{(11),0}(\vec{x}),
\\ \dsp
\Phi_{(12)}(\vec{x})&=&
 \Phi_{(11),1}(\vec{x})\, x_{(2)}
+ \Phi_{(11),2}(\vec{x})\, x_{(3)}
+ \Phi_{(11),3}(\vec{x})\, x_{(4)}
+ \Phi_{(11),4}(\vec{x})\, x_{(12)}\,.
\arre \eqne

\item
\label{Phi44}

 For each $I\not\in{\cal K}_{res}$ there exist positive integers
 $m=m_I$ and $m'=m'_I$
 such that $\Phi_{(1)}$ and $\Phi_{(11)}$ contain terms
 $x_{(1)}^m x_I$ and $x_{(1)}^{m'} x_I$\,, respectively.

\item
\label{Phi45}

 If $I\not\in{\cal K}_{res}$\,, then each term in $\Phi_I$ contains
 a positive power of $x_J$ for some $J\not \in {\cal K}_{res}$\,.
 Furthermore, each term in $\Phi_{(3)}$ and $\Phi_{(4)}$ has
 total degree $2$ or more of $x_J$'s with  $J\not \in {\cal K}_{res}$\,.
 $\Phi_{(2)}$ contains a term $x_{(1)}^3 x_{(11)} x_{(12)}$ and
 $\Phi_{(12)}$ contains a term $x_{(1)}^4 x_{(2)}$\,.

\itme
\prpe
\prfb
\itmb
\item
 This is already proved in \prpu{RG}.

\item
 The explicit forms \eqnu{Phi4}
 of $\Phi_{I}(x,y,0,0,0,0)$, $I=(1), (1,1)$,
 are obtained by explicit counting of paths in the right hand side of
 the definition \eqnu{genfcn} for $\vec{\Phi}=\vec{X}_{1}$\,,
 with aid of computer calculations.

\item

 The expression for $\Phi_{(2)}$ in \eqnu{Phi40}
 follows from an observation (from the definition \eqnu{SAPI})
 that each path $w$ in $W_{(2)}^{(1)}$, which starts from $O$ and
 ends at $v_{1,2}$\,, must pass through $v_{1,1}$\,.
 $w$ therefore has a path segment of length more than $1$
 in the unit simplex $F_0+v_{0,1}$ (which contains $v_{1,1}$).
 The contribution $x_I$ to $\Phi_{(2)}(\vec{x})$ from this simplex is
 therefore in $\{ x_I\mid I\not\in{\cal K}_{res} \} $.
 Classifying the terms in$\Phi_{(2)}(\vec{x})$ by this $x_I$\,,
 we obtain the claimed expression.

 The expression for $\Phi_{(12)}$ follows in a similar way.

 To relate $\Phi_{(2)}$ to $\Phi_{(1)}$\,,
 classify the terms again by the contribution from $F_0+v_{0,1}$\,.
 Let $w\in W^{(1)}_{(2)}$.
 According to what is just proved, $\hat{w}\cap (B_0+v_{0,1})$ is congruent to
 one of $\{\Delta_I \mid I\not\in {\cal K}_{res}\}$,
 in the notation of \prpu{dof}.

 If it is congruent to $\Delta_{(2)}$\,,
 then $w$ has a form $\overline{v_a v_{1,1} v_b}$ in the simplex.
 To this $w$ assign a path $w'$
 obtained by replacing $\overline{v_a v_{1,1} v_b}$
 by $\overline{v_a v_b}$.
 Then $w'\in W^{(1)}_{(1)}$, and contributes a term to $\Phi_{(1)}$
 similar to the contribution of $w$ to $\Phi_{(2)}$ but with $x_{(2)}$
 replaced by $x_{(1)}$\,.

 If $w\cap (B_0+v_{0,1})$ is congruent to $\Delta_{(3)}$\,,
 then $w$ has a form $\overline{v_a v_{1,1} v_c v_b}$ or
 $\overline{v_a v_c v_{1,1} v_b}$ in the simplex.
 To this $w$ assign a path $w'$
 obtained, as before, by shortcutting $v_{1,1}$\,,
 to obtain $\overline{v_a v_c v_b}$\,.
 Then $w'\in W^{(1)}_{(1)}$, and contributes a term to $\Phi_{(1)}$
 similar to the contribution of $w$ to $\Phi_{(2)}$ but with $x_{(3)}$
 replaced by $x_{(2)}$\,.
 This correspondence is $2$ to $1$, which explains the factor $\dsp \frac12$
 in the expression of $\Phi_{(1)}$ in \eqnu{Phi40}.

 The other possibilities $(4)$ and $(12)$ are similar.

 That $x_{(1)}^2$ is contained in $\Phi_{(1)}(\vec{x})$ is
 proved in \prpu{RG}.
 This term can not be obtained by an above mentioned
 `shortcutting' procedure on paths, so it must be contained in
 $\Phi_{(1),0}(\vec{x})$.

 The expression for $\Phi_{(11)}$ follows in a similar way.

\item

 $W^{(1)}_{(1)}$ contains paths
 $\overline{Ov_{1}(v_{1}+v_{2})v_{1,1}}$\,,
 $\overline{Ov_{1}(v_{1}+v_{2})(v_{1}+v_{3})v_{1,1}}$\,,
 $\overline{Ov_{1}(v_{1}+v_{2})(v_{1}+v_{3})(v_{1}+v_{4})v_{1,1}}$\,,
and
 $\overline{Ov_{2}(v_{2}+v_{3}) v_{3} v_{4} v_{1} v_{1,1}}$\,,
 which respectively contribute
 terms $x_{(1)} x_{(i)}$\,, $i=2,3,4$, and $x_{(1)}^3 x_{(12)}$ in
 $\Phi_{(1)}$\,.

 Similarly, paths sets
\[\arrb{l}\dsp
 (\overline{Ov_{1}v_{1,1}}\,,
 \overline{v_{1,2}(v_{3}+v_{2})(v_{3}+v_{1})v_{1,3}}),
\\ \dsp
 (\overline{Ov_{1}v_{1,1}}\,,
 \overline{v_{1,2}(v_{3}+v_{2})(v_{3}+v_{1})(v_{3}+v_{4})v_{1,3}}),
\\ \dsp
 (\overline{Ov_{1}v_{1,1}}\,,
 \overline{v_{1,2}(v_{3}+v_{2})(v_{3}+v_{1})(v_{3}+v_{4}) v_{3} v_{1,3}}),
\\ \dsp
 (\overline{Ov_{1}v_{1,1}}\,,
 \overline{v_{1,2} v_{2} v_{4} v_{3} v_{1,3}}),
\arre \]
 respectively contribute
 terms $x_{(1)}^3 x_{(i)}$\,, $i=2,3,4$, and $x_{(1)}^3 x_{(12)}$ in
 $\Phi_{(11)}$\,.

\item

 The first part follows from \eqnu{Phi40} and similar expressions for
 $\Phi_{(3)}$, $\Phi_{(4)}$\,.
 The second part follows from an observation similar to that used to prove
 \eqnu{Phi40}.
 That $\Phi_{(2)}$ contains a term $x_{(1)}^3 x_{(11)} x_{(12)}$
 follows from an existence of a path
\[
 \overline{O v_{3} (v_{1}+v_{3})(v_{1}+v_{4}) v_{4} v_{1} v_{1,1}
 (v_{1}+v_{2}) v_{1,2}}\,,
 \]
 and $\Phi_{(12)}$ contains a term $x_{(1)}^4 x_{(2)}$
 because of a path
\[
 (\overline{Ov_{1}v_{1,1}}\,,
 \overline{v_{1,2} (v_{2}+v_{3}) v_{1,3} (v_{3}+v_{4}) v_{1,4}}).
\]

\itme
\QED\prfe

\thmb
\thma{4SGrestrict}
\itmb
\item
 $\dsp \Phi_{(1)}(x,y,0,0,0,0) = x$, and
 $\dsp \Phi_{(11)}(x,y,0,0,0,0) = y$ has a unique solution
 $\vec{x}_c=(x_c,y_c,0,0,0,0)$
 in $\dsp \{(x,y,0,0,0,0)\in \preals^{{\cal K}_{res}}\mid (x,y,0,0,0,0)\in
 \Xi_4\setminus \{\vec{0}\}\}$.

 $x_c=0.326490898\cdots$ and
 $y_c=0.027929572\cdots$ are positive.
 (In particular, $\dsp {\cal K}_{res}={\cal K}_{\vec{x}_c}$\,.)

\item
 $\vec{x}_c$ is a self-avoiding fixed point, i.e., satisfies
 (FP\ref{FPbegin}) -- (FP\ref{FPend}).

\item
 There exists a critical point of the restricted model $\beta_{c,res}$\,.

\itme
\thme
\prfb
\itmb
\item

 Since \eqnu{Phi4} implies
 $x> x^2+3x^3+6x^4+6x^5$, it follows that $\dsp 0< x< \frac38$\,.

 Eliminating $y$ from
 $\Phi_{(1)}(x,y,0,0,0,0)=x$ and $\Phi_{(11)}(x,y,0,0,0,0)=y$
 (in a similar spirit with that of
 the proof in \cite[Proposition 3.1]{HHK93},
 but with lengthy calculations),
 we obtain an algebraic equation $g(x)=0$, where
\[ \arrb{rcl}\dsp
g(x)&=&-3162456 + 3162456 x + 27935028 x^2 + 82351390 x^3
 + 534340195 x^4 - 22712313853 x^5
\\ &&\dsp {}
 - 22749190609 x^6 + 173488539516 x^7 + 520491536505 x^8
 + 159919155293 x^9
\\ &&\dsp {}
 - 1067593750255 x^{10} - 3355567112768 x^{11} - 7117707818273 x^{12}
 - 8049744033921 x^{13}
\\ &&\dsp {}
 + 3218074725393 x^{14} + 29132597332920 x^{15} + 58986824992938 x^{16}
 + 74778447132144 x^{17}
\\ &&\dsp {}
 + 70897214418552 x^{18} + 55063893147408 x^{19} + 36096140965140 x^{20}
\\ &&\dsp {}
 + 19669482325692 x^{21}
 + 7841354208804 x^{22} + 1771680351168 x^{23} + 149567809608 x^{24},
\arre \]
 and $y$ is expressed by a rational function of $x$, say $h(x)$.

 The signs of the coefficients imply (by an argument from elementary
 calculus similar to those in the proof of \cite[Proposition 3.1]{HHK93})
 that $g'(x)$ changes sign no more than $4$ times on $x\ge 0$
 (i.e., $g'(x)$ has no more than $4$ positive roots).
 Since, by explicit calculation, $g'(0)>0$, $g'(1/5)<0$, $g'(3/8)>0$,
 $g'(3/7)<0$, and $g'(+\infty)>0$, $g'(x)$
 changes sign $4$ times on $x\ge0$, and $2$ times on $0\le x\le 3/8$.
 Also we have $g(0)<0$, $g(1/5)<0$, and $g(3/8)>0$\,.
 In particular, we have one and only one solution $x=x_c$ of $g(x)=0$
 in $x\in (1/5,3/8)$.

 Again by explicit calculation, we have $g'(21/200)>0$ and $g'(11/100)<0$,
 therefore, the other point that $g'(x)$ changes sign in $0\le x\le 3/8$
 is in the interval $(21/200,11/100)$.
 By explicit calculation $g'(x)\le g'(21/200)\le 9\times 10^5$,
 $x\in[21/200,11/100]$, and $g(21/200)\le -2\times 10^6$.
 Therefore, if $x\in[21/200,11/100]$,
 then $g(x)\le -2\times 10^6+4.5\times 10^3<0$.
 This implies $g(x)<0$, $0\le x\le 1/5$, hence $x_c$ obtained above is
 the unique solution to $g(x)=0$ on $x\ge 0$\,.
 With $y_c=h(x_c)$, $(x_c,y_c)$ is therefore the unique solution to
 $\dsp \Phi_{(1)}(x,y,0,0,0,0) = x$, and
 $\dsp \Phi_{(11)}(x,y,0,0,0,0) = y$ on $\preals^2$.
 It is easy to see that $x_c=0.326490898\cdots$ and
 $y_c=0.027929572\cdots$, which prove that
 $(x_c,y_c,0,0,0,0)\in \Xi_4\setminus \{\vec{0}\}$\,.

\item

 We prove each of (FP\ref{FPbegin}) -- (FP\ref{FPend}) in turn.
\def\labelenumi{(FP\arabic{enumi})}
\itmb
\item

 \prpu{Phi4}~\prpi{Phi45} implies that
 if $I\not\in{\cal K}_{res}=\{(1), (11)\}$
 then $\Phi_{I}(\vec{x}_c)=0$.
 Therefore $\dsp\vec{\Phi}(\vec{x}_c)=\vec{x}_c$\,.

\item

 \prpu{Phi4}~\prpi{Phi45} implies
\[ \Jacobi_{(3)J}(\vec{x}_c)=\Jacobi_{(4)J}(\vec{x}_c)=0,
\ \ \ J\in{\cal I}_d\,, \]
 and
\[ \Jacobi_{I(1)}(\vec{x}_c)= \Jacobi_{I(11)}(\vec{x}_c)=0,
\ \ \ I\not\in{\cal K}_{res}\,. \]
 Therefore, $B=\Jacobi(\vec{x}_c)$ has a form
\[B
=\left(
\begin{array}{cccccc}
p &3q & B_{13} & B_{14} & B_{15} & B_{16} \\
q & r & B_{23} & B_{24} & B_{25} & B_{26} \\
0 & 0 & B_{33} & B_{34} & B_{35} & B_{36} \\
0 & 0 &    0   &    0   &    0   &    0   \\
0 & 0 &    0   &    0   &    0   &    0   \\
0 & 0 & B_{63} & B_{64} & B_{65} & B_{66}
\end{array}
\right)\,,\]
 where, by \eqnu{Phi4},
\[ \arrb{l} \dsp
p = 2x_c + 9x_c^2 + 24x_c^3 + 30x_c^4 + 36x_c^2 y_c + 120x_c^3 y_c
+ 36 x_c y_c^2 + 234x_c^2 y_c^2 + 192x_c y_c^3 + 132 y_c^4\,,
\\ \dsp
q = 4x_c^3 + 10x_c^4 + 12x_c^2 y_c + 52x_c^3 y_c + 96x_c^2y_c^2
+ 176x_c y_c^3 + 220y_c^4\,,
\\ \dsp
r = 4x_c^3 + 13x_c^4 + 64x_c^3 y_c + 264x_c^2 y_c^2 + 88y_c^3
+ 880x_c y_c^3 + 930 y_c^4\,,
\arre \]
 and $B_{ij}$'s are non-negative, and \eqnu{Phi40} implies
\[ \arrb{l} \dsp
B_{33}=\Jacobi_{(2)(2)}(\vec{x}_c)= \Phi_{(1),1}(\vec{x}_c),
\\ \dsp
B_{36}=\Jacobi_{(2)(12)}(\vec{x}_c)= \Phi_{(1),4}(\vec{x}_c),
\\ \dsp
B_{63}=\Jacobi_{(12)(2)}(\vec{x}_c)= \Phi_{(11),1}(\vec{x}_c),
\\ \dsp
B_{66}=\Jacobi_{(12)(12)}(\vec{x}_c)= \Phi_{(11),4}(\vec{x}_c).
\arre \]

 A $2\times 2$ matrix
 $\dsp \left( \begin{array}{cc} p &3q \\ q & r \end{array}\right)$
 has eigenvalues $\lambda=3.17282866849\cdots>1$
 and $\lambda_2=0.249393708\cdots$.
 In fact, they are the solutions of $\dsp x^2-(p+r) x+pr-3q^2=0$\,.

 Consider a $2\times 2$ matrix
 $\dsp B'=\left( \begin{array}{cc} B_{33} & B_{36} \\
 B_{63} & B_{66} \end{array}\right)$.
 Its characteristic polynomial $f(x)=\det (x\,I-B)$ is
\eqnb
\eqna{Bchrpoly}
f(x)=x^2 -(B_{33}+B_{66}) x + (B_{33} B_{66} - B_{36} B_{63})
=(x-\frac12(B_{33}+B_{66}))^2-\frac14(B_{33}-B_{66})^2- B_{36} B_{63}\,.
 \eqne

 Note that \eqnu{Phi40} implies
\[ \arrb{l} \dsp
 x_c=\Phi_{(1)}(\vec{x}_c)
= B_{33}\, x_c + B_{36}\, y_c + \Phi_{(1),0}(\vec{x}_c)
\ge B_{33}\, x_c + B_{36}\, y_c + x_c^2,
\\
 y_c=\Phi_{(11)}(\vec{x}_c)
= B_{63}\, x_c + B_{66}\, y_c + \Phi_{(11),0}(\vec{x}_c)
\ge B_{63}\, x_c + B_{66}\, y_c + x_c^4,
\arre \]
 which further imply
\[ \arrb{l}\dsp
0\le x_a:=
 \frac12(B_{33}+B_{66}) \le \frac12((1-x_c)+(1-\frac{x_c^4}{y_c}))<1,
\\ \dsp
1-B_{33}-B_{66}+B_{33}B_{66}=(1-B_{33})(1-B_{66})
\ge (\frac{y_c}{x_c}B_{36}+x_c)(\frac{x_c}{y_c} B_{63}+\frac{x_c^4}{y_c})
> B_{36}B_{63}\,.
 \arre \]
 Note also that \prpu{Phi4}~\prpi{Phi45} implies that
\[ B_{36}\ge x_c^3 y_c \ \ \mbox{ and }\ \ B_{63}\ge x_c^4. \]

 Using these estimates in \eqnu{Bchrpoly} we find that $f(x)$ is
 increasing on  $\dsp x\ge x_a$,
 symmetric with respect to $\dsp x=x_a$
 (hence in particular, $f(-1)> f(-1+2x_a)=f(1)$),
 $\dsp f(x_a)\le - B_{36} B_{63}\le -x_c^7 y_c<0$,
 and $\dsp f(1)>0$\,.
 Therefore the two eigenvalues $\lambda_3$ and $\lambda_4$
 of $B'$ are real, distinct,
 and have absolute values less than $1$\,.

 The other $2$ eigenvalues are $0$, for which we have
 $2$ independent left eigen vectors $(0,0,0,1,0,0)$ and $(0,0,0,0,1,0)$.

 Therefore, $B=\Jacobi(\vec{x}_c)$ is diagonable by an invertible matrix,
 whose eigenvalues are $\lambda$, $\lambda_2$, $\lambda_3$, $\lambda_4$,
 $0$, $0$,
 and the eigenvalue $\lambda$ which is largest in absolute value satisfies
 $\lambda>1$ and
 all the other eigenvalues have absolute values
 strictly less than $1$\,.

\smallskip\par

 Denote the left eigenvector of $B$ for $\lambda$ by
 $\dsp\vec{v}_L=(v_{L,(1)}, v_{L,(11)}, v_{L,(2)}, v_{L,(3)}, v_{L,(4)},
 v_{L,(12)})$\,.
 It is defined by
\[\arrb{l} \dsp
\frac{v_{L,(11)}}{v_{L,(1)}} = \frac{\lambda-p}{q} = \frac{3q}{\lambda-r}\,,
\\ \dsp
v_{L,(2)}=\frac1{\lambda-B_{33}}
 (B_{13}v_{L,(1)}+ B_{23}v_{L,(11)}+ B_{63}v_{L,(12)}) 
\\ \dsp
v_{L,(3)}=\frac1{\lambda}
 (B_{14}v_{L,(1)}+ B_{24}v_{L,(11)}+ B_{34}v_{L,(2)}+ B_{64}v_{L,(12)}),
\\ \dsp
v_{L,(4)}=\frac1{\lambda}
 (B_{15}v_{L,(1)}+ B_{25}v_{L,(11)}+ B_{35}v_{L,(2)}+ B_{65}v_{L,(12)}),
\\ \dsp
v_{L,(12)}=\frac1{\lambda-B_{66}}
 (B_{16}v_{L,(1)}+ B_{26}v_{L,(11)}+ B_{36}v_{L,(2)}).
\arre \]

 Obviously, we can take $v_{L,(1)}=1$\,.
 $\lambda$ is the larger solution of $\dsp x^2-(p+r) x+pr-3q^2=0$,
 hence $\lambda>p$. Therefore, $v_{L,(11)}>0$\,.
 Note that \prpu{Phi4}~\prpi{Phi44} implies
\[ B_{i,j}>0,\ \  i=1,2,\ j=3,4,5,6\,. \]
 Note also that $\dsp \lambda>1> \max\{B_{33},B_{66}\}$\,.
 Therefore $v_{L,I}>0$, $I=(3),(4),(5),(12)$,

 Finally, note that the right eigenvector of $B$ for $\lambda$
 can be chosen to be
 $\dsp\left(\begin{array}{c} 3q \\ \lambda-p \\0\\0\\0\\0\end{array}\right)$,
 hence has positive (non-zero) $(1)$-component.

\item

 The explicit form \eqnu{Phi4} shows that
 $\Phi_{(1)}$ and $\Phi_{(11)}$ contain terms $x_{(1)}^2$ and $x_{(1)}^4$,
 respectively.

\item
 We have already shown
 $\dsp \vec{x}_c\in\Xi_4\cap \preals^{{\cal K}_{res}}$
 and since $x_c>0$, $\vec{x}_c\ne \vec{0}$.

\itme
\def\labelenumi{(\roman{enumi})}


\item
 \thmu{D} implies that there exists one and only one $\beta_0\in \reals$
 such that $\dsp \vec{x}_{can,{\cal K}_{res}}(\beta_0)\in \partial D$.
 To prove that $\beta_0$ is the critical point of the restricted model,
 it is sufficient to prove, from (CS\ref{CSrestrict}) and
 (DA\ref{DAbegin}) -- (DA\ref{DAend}), that
\eqnb
\eqna{DA14SG}
 \limf{n} \vec{X}_n(\vec{x}_{can,{\cal K}_{res}}(\beta_0))=\vec{x}_c\,,
\eqne
 and
\[ x_{can,{\cal K}_{res},I}(\beta_0)\ne 0,\ I\in{\cal K}_{res}\,. \]
 The latter obviously holds, because
\[ \vec{x}_{can,{\cal K}_{res}}(\beta_0)
 = (e^{-\beta_0}, e^{-2\beta_0}, 0,0,0,0). \]
 Hence it suffices to prove \eqnu{DA14SG}.

 Since $\preals^{{\cal K}_{res}}\subset \preals^{{\cal I}_4}$ is
 an invariant subset (\prpu{restrictedisinvariant} or \prpu{zerocomponent}),
 we may restrict \eqnu{DA14SG} to $\preals^{{\cal K}_{res}}$\,.
 Define
\[ \arrb{l}
 x_n(x,y)= X_{n,(1)}(x,y,0,0,0,0),
\ \ \ y_n(x,y)= X_{n,(11)}(x,y,0,0,0,0),
\\ \dsp
\vec{\phi}=(\phi_1,\phi_2);\ \ \  \phi_1(x,y)=\Phi_{(1)}(x,y,0,0,0,0),
\ \ \phi_2(x,y)=\Phi_{(11)}(x,y,0,0,0,0).
\arre \]
 Then $(x_0(x,y),y_0(x,y))=(x,y)$ and
 $\dsp (x_{n+1},y_{n+1})=\vec{\phi} \circ (x_n,y_n)$, $n\in\pintegers$.

 Next define
\[ \Xi_4^{(2)}=\{(x,y)\in\preals^2\mid (x,y,0,0,0,0)\in \Xi_4\}
=\{(x,y)\in\preals^2\mid x^2\ge y\}, \]
 and
\[ D^{(2)}=\{(x,y)\in\Xi_4^{(2)} \mid (x,y,0,0,0,0)\in D\}, \]
 where $D\subset \preals^6$ is as in \eqnu{D}.
 Denote by $D^{(2)o}$, $D^{(2)c}$, $\partial D^{(2)}$, the
 interior, exterior, and boundary of $D^{(2)}$ in $\Xi_4^{(2)}$.

 Note that
\[ (e^{-\beta_0}, e^{-2\beta_0})\in \partial D^{(2)}. \]
 To prove \eqnu{DA14SG}, it then suffices to prove
\eqnb
\eqna{DA14SG2}
 \limf{n} (x_n(x,y),y_n(x,y))=(x_c,y_c),
\ \ (x,y)\in \partial D^{(2)}.
\eqne

 The definition and \thmu{D} implies that
 these sets are invariant sets of $(\phi_1,\phi_2)$ and that
\[ \arrb{l} \dsp D^{(2)}
=\{(x,y)\in\Xi_4^{(2)} \mid
 \sup_{n\in\pintegers} \max\{ x_n(x,y), y_n(x,y)\} <\infty\}
\\ \dsp
=\{(x,y)\in\Xi_4^{(2)} \mid
 \sup_{n\in\pintegers} \max\{ x_n(x,y), y_n(x,y)\} \le 1 \},
\arre \]
 and
\[ D^{(2)o}
=\{ (x,y)\in\Xi_{4}^{(2)} \mid
\limf{n} \max\{ x_n(x,y), y_n(x,y) \}= 0 \}. \]
 The explicit form \eqnu{Phi4} and \thmu{D} also imply,
 by an elementary argument (as in \cite[Proposition 3.2 (2)]{HHK93}),
 that there exist a constant $c$, satisfying $\dsp 0<c<\frac12$\,,
 and a continuous sitrictly decreasing function $p:\ [0,c]\to\preals$,
 satisfying $\dsp p(0)<\frac12$ and $p(c)=0$,
 such that
\eqnb
\eqna{pD2}
 \partial D^{(2)}=\{(x,p(x))\mid x\in[0,c]\}.
\eqne

 The Jacobi matrix of the map $(x,y)\to \vec{\phi}(x,y)$ is,
 by \eqnu{Phi4},
\[ \arrb{rcl} \dsp
\Jacobi^{(2)}(x,y) &= &
 \dsp \frac{\partial (\phi_1,\phi_2)}{\partial (x,y)}(x,y)
\\ &=& \dsp
   8 x^4 + 62 x^5 + 165 x^6 + 192 x^7 + 90 x^8
 + 128 x^4 y + 432 x^5 y + 516 x^6 y + 360 x^7 y
\\ & & \dsp {}
 + 528 x^3 y^2 + 2088 x^4 y^2 + 3996 x^5 y^2
 + 4770 x^6 y^2 + 176 x y^3 + 2552 x^2 y^3
\\ & & \dsp {}
 + 10032 x^3 y^3
 + 25200 x^4 y^3 + 35040 x^5 y^3 + 1860 x y^4 + 11538 x^2 y^4
 + 56640 x^3 y^4
\\ & & \dsp {}
 + 113520 x^4 y^4 + 3168 x y^5
 + 69912 x^2 y^5 + 206640 x^3 y^5 + 50376 x y^6 + 201780 x^2 y^6
\\ & & \dsp {}
 + 11616 y^7 + 62400 x y^7 - 22440 y^8.
\arre \]

 We note the following fact proved by an elementary argument
 in \cite[Proposition 3.3]{HHK93}.
\lemb
\lema{J2}
 If there exists $\epsilon>0$ such that
\eqnb
\eqna{J2}
 \Jacobi^{(2)}(x,y)>\epsilon,\ (x,y)\in \partial D^{(2)},
\eqne
 then,
 $\dsp \phi_1(x_1,y_1)<\phi_1(x_2,y_2)$ holds for
 any $(x_1,y_1)\in \partial D^{(2)}$ and
 $(x_2,y_2)\in \partial D^{(2)}$ satisfying $x_1<x_2$\,.

\leme
\remb
 The proof of \cite[Proposition 3.3]{HHK93} applies
 to the present case. The proof there
 (after \cite[(3.7)]{HHK93}) uses only \eqnu{pD2} and
 the fact that $\phi_1$ and $\phi_2$
 are polynomials of positive coefficients.
\DDD
\reme

 To prove \eqnu{J2},
 note that the only negative term in $\Jacobi^{(2)}(x,y)$ is 
 $-22440 y^8$.
 If $(x,y)\in \partial D^{(2)}$, then $y=p(x)<p(0)<1/2$, hence
 $22440y^8<11220y^7<11616y^7$.
 With other positive terms in $\Jacobi^{(2)}(x,y)$,
 \eqnu{J2} follows easily.

\smallskip\par

 We are ready to prove \eqnu{DA14SG2}.
 Let $(x,y)\in \partial D^{(2)}$, and assume that
 $\{(x_n(x,y),y_n(x,y))\mid n\in\pintegers\}$ accumulates at
 $(x_a,y_a)\ne (x_c,y_c)$.
 Let $(x_b,y_b)=\vec{\phi}(x_a,y_a)$.
 Since \thmu{4SGrestrict} implies that
 $(x_c,y_c)$ is the only fixed point in $\Xi^{(2)}_4\setminus\{\vec{0}\}$,
 it follows that $x_b\ne x_a$\,.

 Assume that $x_b>x_a$.
 For $\epsilon>0$ denote the $\epsilon$ neighborhood of $(x,y)$
 by $U_{\epsilon}(x,y)$.
 Since $\vec{\phi}$ is continuous, for any $\epsilon>0$ there exists
 $\delta>0$ such that
 $\dsp \vec{\phi}(U_{\delta}(x_a,y_a))\subset U_{\epsilon}(x_b,y_b)$.
 Taking $\epsilon$ small enough so that
 $(x',y')\in U_{\epsilon}(x_b,y_b)$ implies
 $x'> x_m:= (x_a+x_b)/2$,
 we therefore see that there exists $\delta>0$ such that
 $(x,y)\in U_{\delta}(x_a,y_a)\cap \partial D^{(2)}$ implies
 $\phi_1(x,y)> x_m$ $( > x_a)$.
 Then \lemu{J2} further implies, by induction,
\[ \phi_1 (\vec{\phi}^{n}(x,y))> x_m\,,
\ \ n\in\nintegers,\ (x,y)\in U_{\epsilon}(x_a,y_a)\cap \partial D^{(2)}. \]

 By assumption that
 $\{(x_n(x,y),y_n(x,y))\mid n\in\pintegers\}$ accumulates at
 $(x_a,y_a)$,
 there exists a positive integer $N$ such that
 $(x_N(x,y),y_N(x,y))\in U_{\epsilon}(x_a,y_a)\cap \partial D^{(2)}$.
 Therefore 
 $\dsp x_n(x,y) > x_m$\,, $n=N+1,N+2,\cdots$, which contradicts the
 assumption that  
 $\{x_n(x,y) \mid n\in\pintegers\}$ accumulates at $x_a<x_m$\,.

 Similar argument also holds for if we assume $x_b<x_a$.
 Therefore $(x_c,y_c)$ is the only point that
 $\{(x_n(x,y),y_n(x,y))\mid n\in\pintegers\}$ accumulates,
 which proves \eqnu{DA14SG2}.

\itme

\QED\prfe

\bigskip\par

 \thmu{4SGrestrict} and the results in \secu{mainresults}
 imply the following results on the asymptotic behaviors of
 restricted self-avoiding paths on the $4$ dimensional pre-\sg.
\thmb
 Let $\vec{x}_c=(x_c,y_c,0,0,0,0)\in\preals^6$ and
 $\beta_{res}=\beta_{c,{\cal K}_{res}}\in\reals$ be
 the constants defined in \thmu{4SGrestrict}.
 Then the following holds for the
 restricted self-avoiding paths on the $4$ dimensional pre-\sg.

\itmb
\item
 If $\vec{x}\in\Dom (\vec{x}_c)$, then the following hold.

 For ${I}\in {\cal K}_{res}=\{(1), (11)\}$,
 the joint distribution of scaled generalized path length
 $(\lambda^{-n} {s}_{(1)}\,,\; \lambda^{-n} {s}_{(11)})$
 under
 $\dsp {\mu}_{\vec{x},n,{I}}$
 converges weakly to a Borel probability measure
 $\dsp p^*_{\vec{x},{I}}$
 on $\dsp\reals^6$ as $n \to \infty$.

 The generating function $\dsp {\varphi}^*_{\vec{x},{I}}$,
 defined by
\[ {\varphi}^*_{\vec{x},{I}}(\vec{{t}})
= \int_0^{\infty} e^{\vec{{t}}\cdot \vec{{\xi}}}
 p^*_{\vec{x},{I}}[d\vec{{\xi}}],
\ \ \ \vec{{t}}\in\cmplx^6\,,
\]
 is an entire function in
 $\dsp\vec{{t}}$, and
 the set of functions
 $(\varphi^*_{\vec{x},(1)}\,,\; \varphi^*_{\vec{x},(11)})$
 are uniquely determined by
 \eqnu{definingchrfcn} for $d=4$\,.

 If $\vec{x}\in \Dom(\vec{x}_c)\cap \Xi_4$
 and $I\in{\cal K}_{res}$\,,
 then
 the distribution of $\dsp\lambda^{-n} L(w)$,
 the scaled length of $\dsp w\in W^{(n)}_{I}$,
 under $\dsp \mu_{\vec{x},n,I}$ converges weakly
 to a Borel probability measure
 $\dsp\bar{p}^*_{\vec{x},I}$\,,
 which has a $C^{\infty}$ density
 $\dsp \bar{\rho}^*_{\vec{x},I}$\,.

 In particular,
 $\dsp \bar{\rho}^*_{\vec{x},{(1)}}(\xi)>0$, $\xi>0$\,. 

\item
 For $I\in{\cal I}_4=\{(1), (11), (2), (3), (4), (12)\}$\,,
\[ \limf{n} Z_{{\cal K}_{res},n,I}(\beta)= \left\{\arrb{ll}\dsp
0, & \beta>\beta_{res}\,,
\\ \dsp
x_{c,I}\,, & \beta=\beta_{res}\,,
\arre\right. \]
 and
\[ \limf{n} \sum_{i=1}^4 Z_{{\cal K}_{res},n,(i)}(\beta)= \infty,
\ \ \  \beta<\beta_{res}\,. \]

\item
 The number $N_{res}(k)=N_{{\cal K}_{res}}(k)$
 of restricted self-avoiding paths of length $k$ starting
 from $0$ satisfies
\[ \limf{k} \frac1k \,\log N_{res}(k) = \beta_{res}\,. \]

\item
 The exponent for mean square displacement for the restricted model
 is $\dsp d_w=\frac{\log \lambda}{\log w}=1.6657696\cdots$,
 in the sense that
\[ \limf{k} \frac1{\log k} \log E_{res,k}[|w(k)|^{s\, d_w}]=s,
\ \ s\ge 0, \]
 where $E_{res,k}$ is the expectation with respect to the probability
 measure with equal weight on length $k$ restricted self-avoiding paths
 starting at $O$.

\itme
\thme
%
\remb
 The convergence of $\dsp {\mu}_{\vec{x},n,{I}}$ and the properties
 of the limit measure holds both for the full model and the restricted
 model (if $\vec{x}\in\Dom (\vec{x}_c)$),
 because they hold independently of (CS\ref{CSrestrict}).

\DDD\reme



%-----------------------------------------------------
\begin{thebibliography}{99}
%-----------------------------------------------------

\bibitem{Dhar} D.~Dhar,
\textit{Self-avoiding Random Walks:  Some Exactly Soluble Cases,}
Journ.\ Math.\ Phys.\ \textbf{19} (1978) 5--11.

\bibitem{HS} T.~Hara, G.~Slade,
\textit{Self-avoiding walk in five or more dimensions, \Rom{1},
 the critical behaviour,}
Commun.\ Math.\ Phys.\ \textbf{147} (1992) 101--136.

\bibitem{HS95} T.~Hara, G.~Slade,
\textit{The self-avoiding-walk and percolation critical points
 in high dimensions,}
Combinatrics, Probability and Computing \textbf{4} (1995) 197--215.

\bibitem{HH91} K.~Hattori, T.~Hattori,
\textit{Self-avoiding process on the Sierpi\'{n}ski gasket,}
Probability Theory and Related Fields \textbf{88} (1991) 405--428.

\bibitem{HHK90} K.~Hattori, T.~Hattori, S.~Kusuoka,
\textit{Self-avoiding paths on the pre-Sierpi\'{n}ski gasket,}
Probability Theory and Related Fields \textbf{84} (1990) 1--26.

\bibitem{HHK93} K.~Hattori, T.~Hattori, S.~Kusuoka,
\textit{Self-avoiding paths on the three dimensional Sierpi\'{n}ski gasket,}
Publications of RIMS \textbf{29} (1993) 455--509.

\bibitem{HK92} T.~Hattori, S.~Kusuoka,
\textit{The exponent for mean square displacement of self-avoiding
random walk on Sierpi\'{n}ski gasket,}
Probability Theory and Related Fields \textbf{93} (1992) 273--284.

\bibitem{Irwin} M.~C.~Irwin,
\textit{Smooth Dynamical Systems},
Academic Press, London, 1980.

\bibitem{MS} N.~Madras, G.~Slade,
\textit{The self-avoiding walk,}
Birkh\"auser, 1993.

\end{thebibliography}

\end {document}
\end{document}
---------------0110172331587--
