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dimer, expansion, entropy
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\begin{document}

\title{ Higher order expansions for the entropy of a dimer or a monomer-dimer
system on $d$-dimensional lattices}

\author{P. Butera}
\email{paolo.butera@mib.infn.it}
\affiliation
{Dipartimento di Fisica Universita' di Milano-Bicocca\\
and\\
Istituto Nazionale di Fisica Nucleare \\
Sezione di Milano-Bicocca\\
 3 Piazza della Scienza, 20126 Milano, Italy}
\author{P. Federbush}
\email{pfed@umich.edu}
\affiliation
{Department of Mathematics\\
University of Michigan \\
Ann Arbor, MI 48109-1043, USA\\}
\author{M. Pernici} 
\email{mario.pernici@mi.infn.it}
\affiliation
{Istituto Nazionale di Fisica Nucleare \\
Sezione di Milano\\
 16 Via Celoria, 20133 Milano, Italy}

\date{\today}

\begin{abstract}

  Recently an expansion as a power series in $1/d$ has been presented
  for the specific entropy of a complete dimer covering of a
  $d$-dimensional hypercubic lattice.  This paper extends from 3 to 10
  the number of terms known in the series.  Likewise an expansion for
  the dimer-density $p$-dependent entropy of a monomer-dimer system
  involving a sum $\sum_k a_k(d) p^k$ has been recently offered.  We
  herein extend the number of the known expansion coefficients from 6
  to 20 for the hypercubic lattices of general dimensionality $d$ and from
  6 to 24 for the lattices of dimensionalities $d < 5 $. 
  We show that this extension can lead to accurate numerical estimates
  of the $p$-dependent entropy for lattices with dimension $d > 2$.  
  The computations of this paper have led us to make the following marvelous
  conjecture: {\it In the case of the hypercubic lattices, all the
  expansion coefficients, $ a_k(d) $, are positive! } This paper
  results from a simple melding of two disparate research programs:
  one computing to high orders the Mayer series coefficients of a
  dimer gas, the other studying the development of entropy from these
  coefficients.  An effort is made to make the paper self-contained by
  including a review of the earlier works.


\end{abstract}
\pacs{ 03.70.+k, 05.50.+q, 64.60.De, 75.10.Hk, 64.70.F-, 64.10.+h}
\keywords{Dimer problem }
\maketitle
\section{Introduction and results}
The dimer problem arose in a thermodynamic study of diatomic molecules
and was abstracted into one of the most basic and natural problems in
both statistical mechanics \cite{10, 11,19} 
and combinatorial mathematics\cite{fla}.
In more recent years, dimers found interesting applications also in
information \cite{7} and string theories\cite{IQ2,IQ3}. 

 Given a hyper-simple-cubic (hsc) lattice with number of sites $N$ 
in $d$ dimensions, the dimer problem loosely speaking is to count the
number of different ways dimers (dominoes) may be laid down in the
lattice (without overlapping) to completely cover it. Each dimer
covers two nearest neighbor vertices. It is known that the number of
such coverings is roughly $\exp (\lambda_d N$) for some constant
$\lambda_d$ as $N$ goes to infinity.  In 1980 H.Minc\cite{1} gave a
proof of the asymptotic relation (asymptotic as $d \to \infty$)
\begin{equation}
\lambda_d  \sim \frac{1} {2} {\rm ln}(2d) - \frac{1} {2}
\end{equation} 
In a series of papers\cite{2,3,4,14}, one of the authors, P.F.,
found a mathematical argument for a full asymptotic expansion
\begin{equation}
\lambda_d  \sim \frac{1} {2} {\rm ln}(2d) - \frac{1} {2}+ \frac{c_1} {d} + \frac{c_2} {d^2}+\cdots
\label{1sud}
\end{equation} 
and computed the first three terms in the Table \ref{tab1} also making
the conjecture that no further terms would be computed. He was very
wrong! One of the results of the present paper is the set of
coefficients from $c_4$ to $c_{10}$ reported in Table \ref{tab1}.
\begin{table}[ht]
  \caption{ Expansion coefficients $c_n$   of the dimer entropy 
    $\lambda_d  \sim \frac{1} {2} {\rm ln}(2d) - \frac{1} {2}
    + \sum_n\frac{c_n} {d^n}$
    in the case of the  hyper-simple-cubic lattices. }
\begin{tabular}{|c|c|c|c|}
 \hline
$c_1$&  1/8& $c_6$&20815/21504 \\
$c_ 2$& 5/96  & $c_7$&9151/6144\\
$c_ 3$&  5/64 & $c_8$& 39593/73728 \\
$c_ 4$&237/1280 & $c_9$&-645691/61440\\
$c_5$&349/768  &$c_{10}$&-107753037/901120\\
 \hline
\end{tabular} 
\label{tab1}
\end{table}
Viewing the sequence of $c_i$, we are certainly led to expect the sum
in Eq.(2) to be asymptotic and not convergent.

If we consider covering by dimers of a fraction of the vertices 
 denoted here by $p=2\rho$ (where $\rho$ is the dimer density per site and 
the vertices not covered by dimers are considered covered by monomers)
and as above study the number of such
coverings, we arrive similarly at a function $\lambda_d(p)$ where
\begin{equation}
 \lambda_d(1)=\lambda_d
\label{lasp}
\end{equation} 
Another common notation for $\lambda_d$ is $\tilde{h}_d$. One also studies
\begin{equation}
h_d=\max_{0 \le p \le1} \lambda_d(p)
\end{equation}
For $\lambda_d(p)$ Friedland et al.\cite{6,7} proved the asymptotic relation
(asymptotic as $d \to \infty$) 
\begin{equation}
 \lambda_d(p)=\frac{1} {2} ( p{\rm ln}(2d)-p{\rm ln}(p)-2(1-p){\rm ln}(1-p)-p)
\label{lasp1}
\end{equation} 
Both this equation and Eq.(1) may be viewed as the mean field
approximations for the respective quantities. This was first mentioned
in Ref.[\onlinecite{14}] and is briefly discussed at the end of Section
III.  By a similar development to that in Ref.[\onlinecite{14}] one of
the authors, P.F. and Friedland \cite{13} argued for an expansion
\begin{equation}
 \lambda_d(p)=\frac{1} {2}(p{\rm ln}(2d)-p{\rm ln}(p)-2(1-p){\rm ln}(1-p)-p)
+\sum_{k=2} a_k(d)p^k
\label{laspd1}
\end{equation} 
where, setting $x(d)= \frac{1}{2 d}$, those authors computed the
following coefficients,

$a_2(d)= \frac{1}{4} x$

$a_3(d)= \frac{1}{12} x^{2}$

$a_4(d)= \frac{1}{24} x^{2} \left(- 5 x + 3\right)$

$a_5(d)= \frac{1}{40} x^{3} \left(- 39 x + 20\right)$

$a_6(d)= \frac{1}{60} x^{3} \left(- 19 x^{2} - 30 x + 20\right)$


The main result of this paper is the extension of known values:

$    a_{7}(d)= \frac{1}{84} x^{4} \left(1093 x^{2} - 1008 x + 231\right)$

$    a_{8}(d)= \frac{1}{112} x^{4} \left(967 x^{3} - 35 x^{2} - 602 x + 189\right)$

$    a_{9}(d)= \frac{1}{144} x^{5} \left(- 66047 x^{3} + 68712 x^{2} - 23556 x + 2856\right)$

$    a_{10}(d)= \frac{1}{180} x^{5} \left(- 67721 x^{4} + 18495 x^{3} + 29565 x^{2} - 15405 x + 2232\right)$

$a_{11}(d)= \frac{1}{220} x^{6} \left(5456221 x^{4} - 6452710 x^{3} + 2752860 x^{2} - 524700 x + 39710\right)$



$a_{12}(d)= \frac{1}{264} x^{6} \left(887437 x^{5} + 2477970 x^{4} - 3847316 x^{3} + 1824724 x^{2} - 378004 x + 31130\right)$


$a_{13}(d)= \frac{1}{312} x^{7} (- 614279535 x^{5} + 794742624 x^{4} -
392705664 x^{3} + 95702984 x^{2}$ \\$
 - 11868441 x + 621504)$


$ a_{14}(d)= \frac{1}{364} x^{7} (678357525 x^{6} - 1192936836 x^{5} + 869146005 x^{4} - 339116960 x^{3} + $\\$
75444460 x^{2} - 9220393 x + 497016)$



$a_{15}(d)= \frac{1}{420} x^{8} (89365899701 x^{6} - 124219633888 x^{5} + 68478916835 x^{4} - 19687487260 x^{3} + $\\$
3185117250 x^{2} - 281248772 x + 10870055)$




$a_{16}(d)= \frac{1}{480} x^{8} (- 206929670185 x^{7} + 330409603725 x^{6} - 221634792330 x^{5} + 83075676915 x^{4} - $\\$
19146441210 x^{3} + 2751382878 x^{2} - 231206020 x + 8907885)$



$a_{17}(d)= \frac{1}{544} x^{9} (- 16388790941183 x^{7} + 24197151077904 x^{6} 
- 14547689415128 x^{5} + $\\$ 
4724677127184 x^{4} - 911997832372 x^{3} + 106422324240 x^{2} 
- 7073226040 x + 210678416)$




    $a_{18}(d)= \frac{1}{612} x^{9} $($55311212276891 x^{8} - 89669360611981 x^{7} 
+ 61471303146642 x^{6} $\\$
 - 23833002227449 x^{5} + 5824219780656 x^{4} - 933123781978 x^{3} 
+ 97025317251 x^{2} - 6063514389 x + $\\$ 
176829104)$




$a_{19}(d)= \frac{1}{684} x^{10} (3770925296332945 x^{8} - 5844092886538362 x^{7} + 3760855236979965 x^{6}$\\$ - 1340101438257204 x^{5}
 + 293876531465913 x^{4} - 41181769780866 x^{3} + 
3649368222699 x^{2}$\\$ - 189574974180 x + 4489042410)$



$a_{20}(d)= \frac{1}{760} x^{10} (- 16045042327489089 x^{9} + 26850617367263509 x^{8} - 19173445082939896 x^{7} + $\\$ 
7825625528101485 x^{6} - 2044727194575071 x^{5} + 359651992720132 x^{4} -
43125672212794 x^{3} $\\$ + 3440152700645 x^{2} - 167626520550 x + 3849436062 )$


For the hsc lattices of dimensionality $d < 5$ four more coefficients
$a_{k}(d)$ are available. They are listed in Table \ref{tab2}.
\begin{table}[ht]
\tiny
\caption{ Higher order expansion  coefficients $a_k(d)$ of the dimer entropy 
$\lambda_d(p)$ on the hsc lattices of dimension $d<5$. }
\begin{tabular}{|c l|c l|c l|}
 \hline
$a_{21}(2)$=&255640084561/923589767331840 & $a_{21}(3)$=
&66223472491867/1023724363217633280 &$a_{21}(4)$=
&15299547547784641/968454063869751459840   \\
$a_{22}(2)$=&50273131919/193514046488576 &
 $a_{22}(3)$=&1171503630290797/20269742391709138944 &
$a_{22}(4)$=&117431629955187175/8522395762053812846592 \\
$a_{23}(2)$=&4312434281365/17803292276948992& 
$a_{23}(3)$=&6903357438819689/133201164288374341632 &
$a_{23}(4)$=& 902716034982108733/74672420010376264941568  \\
$a_{24}(2)$=&5789230773063/25895697857380352 & $a_{24}(3)$=
&40662370356724697/871862166251177508864 
&$a_{24}(4)$=&6949151047607061613/651686574636011039490048 \\
 \hline
\end{tabular} 
\label{tab2}
\end{table}
\normalsize
In Ref.[\onlinecite{13}] it was conjectured that the 
series in Eq.(\ref{laspd1}) is
convergent for $0 \le p \le 1 $.  Author P.F. in fact proved\cite{17}
that this series converges for small enough $p$.

We have checked that the $a_k(d)$ are positive for integer values of
$d$ with $ k \le 20$. Due to this and to the behavior of the roots of
$a_k(d)$ (see Appendix), we are then led to make the conjecture that
{\it the $a_k(d)$ are all positive for integer values of $d$ in the
  case of the hsc lattices}.

It follows that the partial sums $\sum_{k \geq 2}^r a_k$ are positive
for integer values of $d$ for any $r$, so that the expansion
Eq.(\ref{laspd1}) gives good approximations of $\lambda_d(1)$ also in
low dimensions, unlike the expansion Eq.(\ref{1sud}), which is
numerically useful only for sufficiently large $d$.  In the Appendix
we shall further discuss this conjecture and shall return to the
numerical approximations in Section IV.


It is interesting to
point out some results of historic importance for the dimer problem.
The exact value of $\lambda_2$  calculated by M.E. Fisher\cite{10}
 and P.W. Kasteleyn\cite{11} is given by the closed form expression
\begin{equation}
  \lambda_{2}\equiv \tilde h_2=\frac{1}{\pi}( \frac{1}{1^2}
- \frac{1}{3^2}+ \frac{1}{5^2}- \frac{1}{7^2} \cdots)= G/\pi= 0.2915609040...
\label{lad3}
\end{equation} 
 with $G$  Catalan's constant.
The technique used in the proof of this relation had great influence
 in the field of exactly soluble models.  

The one-dimensional problem
 has an even more complete solution\cite{6}
\begin{equation}
\lambda_{1}(p)=\frac{p} {2}  {\rm ln}(2)-
\frac{p}{2}{\rm ln}(p)-(1-p){\rm ln}(1-p)
-\frac{p}{2}+\sum_{k=2}^{\infty} \frac{(p/2)^k}{(k-1)k}
\label{lad1}
\end{equation}
 so that $\lambda_1(1)\equiv \tilde h_1=0$ and $h_1={\rm ln} \frac
{1+\sqrt 5}{2}$.
Notice that in this simple case all the $a_k(1) $ are positive and
rapidly vanishing as $k \to \infty$ so that the series converges for $0
\leq p \leq 1$.  

Let us turn for the moment to consideration of a dimer
gas on our $d$ -dimensional lattice.  The gas of dimers is taken as a
``hard body'' system. Between each two dimers there is a potential
energy $0$ if the dimers are disjoint and $+\infty$ if they
overlap. For this gas we are interested in the coefficients in the
Mayer series\cite{ruelle} $ b_1(d), b_2(d),..$.

Both the formalism in Ref.[\onlinecite{14}] used to derive Eq.(2) and
the formalism in Ref.[\onlinecite{13}] used to derive
Eq.(\ref{laspd1}) take as inputs the $b_i(d)$ and have as outputs the
$c_i$ of Table \ref{tab1} and the $a_k(d)$.  Author P.F. did not have
as good an algorithm for computing to high orders the $b_i(d)$ as in
Ref.[\onlinecite{bp1,bp2,bp3}] and was not aware of the
already existing lower-order expansions\cite{gaunt,kurfis,kenzie} for small
lattice dimensionalities.  This explains the many additional terms
computed in Eq.(2) and Eq.(\ref{laspd1}) when the computations of
Ref.[\onlinecite{bp3}] were used as inputs.  In Sect.II, the technique
used in Ref.[\onlinecite{bp2,bp3}] to compute the $b_i(d)$ is
discussed. For the computations of the $ a_i(d)$, $ i = 1,2,...20$,
one needed exactly the $b_i(d)$ for $1 \le i \le 20$, and $1\le d \le
10$.  (Interestingly these values in fact determine the $b_i(d)$, $1
\le i \le 20$, for all $d$.  This will be shown 
in  Section II and in an independent way in Section III.)

In Sect.III the machines in Refs.[\onlinecite{14}]  and
[\onlinecite{13}] to calculate the $c_i$ and $a_k(d)$ respectively, are
discussed. But they are too technical to get deeply into all of the
theory. Recently, in fact within the past year, P.F. found another
route from the $b_i(d)$ to expansions for $\lambda_{d}(p)$, simple
enough for us to completely describe it in this paper\cite{12}.
We close this Section by specializing\cite{15} the expansion in
Eq.(\ref{laspd1}) to $d=2$, to see what such an expansion looks like 
\begin{equation}
 \lambda_2(p) \sim \frac{1} {2}(p{\rm ln}(4)-p{\rm ln}(p)-2(1-p){\rm ln}(1-p)-p)
+ 2(\frac{1}{2\cdot 1}(\frac {p} {4})^2+\frac{1}{3\cdot 2}(\frac {p} {4})^3+
\frac{7}{4\cdot 3}(\frac {p} {4})^4+ \frac{41}{5\cdot 4}(\frac {p} {4})^5\cdots
\label{lad2}
\end{equation} 
where this equation is determined by an infinite sequence of integers
\begin{equation}
1,1,7,41,181,757,3291,14689,64771,276101,1132693,4490513,17337685,...
\label{square}
\end{equation}
of which the first 23 integers are known from the calculations of this
paper.  It is very natural to try to find a pattern in the successive
terms of this sequence so that a closed form expression for
$\lambda_2(p)$ be realized, recalling that it exists for $\lambda_2
= \lambda_2(1)$. 

Recently we came across an early paper by Rushbrooke, Scoins and
Wakefield [\onlinecite{rush}] computing by a somewhat different method
the first six coefficients in Eq.(\ref{square}) for the square lattice
and the first five for other two-dimensional and three-dimensional
lattices.

The rest of the paper is organized as follows.  In Section II we
recall how the Mayer expansion for the dimer problem is related to the
high-temperature low-field expansion of an Ising system.  Section III
sketches how the expansion of Eq.(\ref{laspd1}) is derived from the
Mayer series.  In Section IV we show how simply the expansion
Eq.(\ref{laspd1}) can lead to accurate estimates of the $p$-dependent
entropy $\lambda_d(p)$.  The Appendix contains additional comments on
the positivity conjecture and lists of the coefficients appearing in
some generalizations of Eq.(\ref{laspd1}) to lattices other than the
hsc. We have included in the Appendix also a subsection on the
graphical expansion procedure for the Ising model.

\section{Dimers and the Ising model}
It has long been known\cite{10,11,gaunt,kurfis} that the number of
ways to place $s$ hard dimers onto a lattice can be evaluated up to a
large $s$ by computing, to the same order $s$ and on the same lattice,
the high-temperature (HT) and low-field series expansion of the
free-energy of a spin-$1/2$ Ising model in the presence of a uniform
magnetic field.


The dimer combinatorial problem can be simply formulated in the
language of statistical mechanics. A set of dimers on a $N$-site lattice
($N$ even) is described as a lattice-gas of molecules occupying
nearest-neighbor sites subject to a non-overlap constraint, in terms
of a macrocanonical partition function
\begin{equation} \Xi_N(z)=1+
\sum_{s=1}^{N/2} Z_sz^s= 1+
\sum_{s=1}^{N/2} g_N(s)z^s.
 \label{mdmpfNZ}
\end{equation}
Due to the non-overlap constraint, $Z_s$, the canonical partition
function for a fixed number $s$ of dimers simply counts the allowed
dimer configurations, so that $g_N(s)$ is precisely the number of ways
of placing $s$ dimers over the links of the lattice, and
$z=\exp{(\beta\mu)}$ is the dimer activity. The chemical potential $\mu$,
namely the energy cost of adding one more dimer to the system, is zero
whenever there is room on the lattice for adding one more dimer and
infinite otherwise.  Therefore the value of $\beta=1/k_BT$ with $T$
the temperature and $k_B$ the Boltzmann constant, is irrelevant and
can be fixed to unity. Thus $z=1$ is the value of the activity
describing the combinatorics of a monomer-dimer system i.e. of a dimer
system that does not cover completely the lattice, while $z=\infty$
describes the complete coverings.

  In the
$N \to \infty$ (thermodynamical) limit one gets
\begin{equation}
\Xi(z)=\lim_{N\rightarrow \infty}[\Xi_N(z)]^{1/N}
=1+  \sum_{s=1}^{\infty} g(s)z^s
\label{mdmpf}
\end{equation}
from which a ``pressure" (or macrocanonical potential) can be defined
in the usual way
\begin{equation}
 P(z)={\rm ln}(\Xi(z))
= \sum_{s=1}^{\infty}b_sz^s.
\label{mpot}
\end{equation}
since $\beta=1$.
The dimer density per site $\rho$ is expressed in terms of the pressure by
\begin{equation}
\rho(z)=z \frac{dP} {dz}
= \sum_{s=1}^{\infty}sb_sz^s.
\label{mden}
\end{equation}
The series for $\rho(z)$ can be inverted to get $z$ as a power series
in the density and by substituting $z=z(\rho)$ in Eq. (\ref{mpot}),
$P$ can be expressed as a power series in the density $\rho$, thus
obtaining the virial expansion.  Eqs.(\ref{mpot}) and (\ref{mden}) are
called the Mayer expansions of the dimer lattice-gas.

The specific entropy $s_d(p)$ of a dimer system of density $\rho$ in
$d$ dimensions is
\begin{equation}
s_d(p)/k_B \equiv \lambda_d(p)=-\rho_d(z) {\rm ln} z +P_d(z)=
\frac{1} {2}(p{\rm ln}(2d)-p{\rm ln}p) +O(p)
\label{mentr}
\end{equation}
where the last expression arises by setting $z=z(p)$, and $\rho=p/2$, and observing
that on the hsc lattices $z=\frac{p}{2d}+O(p^2)$.
This structure was further specified in Ref.[\onlinecite{6,7}], as indicated in
Eq.(\ref{3.9}).  
 One can also easily check that changing the variable from $z$ to $p$,
the point $z=1$ corresponds to a stationary point of the entropy with
respect to $p$, thus linking the definition given above of $h_d$ in
terms of $\lambda_d(p)$ with the definition used in
Ref.[\onlinecite{bp1}] as $P(z)|_{z=1}$.
We now couple the relation Eq.(\ref{mentr}) with the expansions above
for $P(p)$ and $z(p)$. We write 
\begin{equation}
z=\frac{p}{2b_1} (1+F(p))
\label{3.6}
\end{equation}
and then get from Eq. (\ref{mentr}) and (\ref{3.6})
\begin{equation}
\lambda_d(p)=P(p)-\frac{p}{2}{\rm ln}(\frac{p}{2b_1})
-\frac{p}{2}{\rm ln}
 (1+F(p))
\label{3.7}
\end{equation}
or 
\begin{equation}
\lambda_d(p)=P(p)-\frac{p}{2}{\rm ln}(p)+ \frac{p}{2}{\rm ln}(2d) -\frac{p}{2}{\rm ln}(1+F(p))
\label{3.8}
\end{equation}
 using $b_1=d$. 
 Referring to Eq.({\ref{laspd1}}) we may put Eq. (\ref{3.8}) in the form
\begin{equation}
\lambda_d(p)=\frac{1}{2}(p{\rm ln}(2d)-p{\rm ln}p -2(1-p){\rm ln}(1-p)-p)+
\sum_{k=2}^{\infty}a_kp^k
\label{3.9}
\end{equation}
when the $a_k$ are suitably determined from the Mayer series coefficients in a
straightforward manner.  


The Mayer coefficients $b_s(d)$ are simply obtained from the HT
expansion of the free-energy for the Ising model.  To illustrate the
relationship between the Ising and the dimer problems, recall the
``primitive''\cite{domb} method of HT and low-field graphical
expansion for the partition function $Z_N(\beta,h)=\sum_{m \ge
  0}\sum_{l=m}^{L_{max}}\gamma_N(2m,l){\rm tanh}(h)^{2m}{\rm
  tanh}(\beta)^l $ of a spin-1/2 Ising model on a lattice of $N$
sites. Here $\beta=1/k_BT$ denotes the inverse temperature and
$h=\beta H$ with $H$ the uniform external magnetic field.  The
expansion coefficient $\gamma_N(2s,s)$ counts all possible lattice
configurations of graphs represented by precisely $s$ disconnected
edges placed onto disjoint links of the lattice and therefore
coincides with the quantity $g_N(s)$ in Eq.(\ref{mdmpf}). The
procedure of forming the specific free-energy
$f_N(\beta,h)=\frac{1}{N}{\rm ln} Z_N$ and then taking the
thermodynamical limit, exactly parallels\cite{kurfis} the procedure
leading to Eq.(\ref{mpot}), so that one concludes that from the
expansion $f(\beta,h)=\sum_{m \ge 0}\sum_{l=m}^{L_{max}}f_{2m,l}{\rm
  tanh}(h)^{2m} {\rm tanh}(\beta)^l$, the Mayer expansion coefficients
can be read as $b_s=f_{2s,s}$.

Let us now recall that recently a significant extension, of the HT
series for several models in the Ising universality class, including
the conventional spin-$1/2$ model, has been obtained for a sequence of
bipartite lattices, in particular the hsc lattices of spatial
dimensionality $1\le d \le 10$ and the hyper-body-centered-cubic
(hbcc) of any dimensionality (at least in principle, because the only
way in which dimension enters is in the power of the embedding
number, and the time it takes to preform
the computation for the  hbcc lattices varies very slowly with the dimension; 
in practice only results for $d \leq 7$ have been obtained).
It is also convenient, at this point, to give some simple details on these
calculations. It is most convenient to use\cite{bp2,bp3} the classical
linked-cluster method\cite{wortis} of graphical expansion.  At each HT
order $l$, the series coefficients are expressed as the sum of an
appropriate class of $l$-edge graphs. Each graph contributes a ratio
of two integers: the ``free-embedding-number'' and the symmetry-number
of the graph, times a product of ``bare vertex-functions'' associated
to the vertices of the graph and depending on the magnetic field. The
embedding-number counts the number of distinct ways (per site of the
underlying lattice) in which the graph can be placed onto the lattice
with each vertex assigned to a site and each edge to a link. This
number depends on the topology of the graph and on the dimensionality
$d$ of the lattice. The important property is that, in the case of the
hsc lattices (but not for the hbcc lattices!), for a generic graph with
$l$ edges, the embedding-number is a polynomial in $d$ of degree $l$
at most.  The symmetry-number counts the automorphisms of the graph
and depends only on the topology of the graph.  The great advantage of
the linked-cluster method comes from the recognition that the huge
variety of graphs that contribute at relatively high orders of
expansion to the computation of a physical quantity, e.g. the
magnetization, can be obtained by combining simpler graphs in a
smaller class\cite{wortis}, thus making possible to trade the
computational complexity for algebraic complexity.

From the field-dependent free-energy, one can compute all its
field-derivatives usually called (higher) susceptibilities. It is
clear at this point that, on the hsc lattices, the computation of
these quantities through the 10th order, can be extended to a generic
$d$, by a simple interpolation of the series coefficients using the
computation on a sequence of hsc lattices of dimensionalities $1 \le
d \le 10$ and based on the fact that the $l$th order expansion
coefficient is a simple polynomial\cite{fisga} of degree $l$ in $d$
(with zero constant term).  Actually much more than this can be
done. One can observe\cite{bp3} that the knowledge of the free-energy
gives access to the HT expansions of the successive derivatives of the
magnetic field with respect to the magnetization $\partial^{2p+1}
h/\partial {\cal M}^{2p+1}$, for $p=0,1...$ and that these quantities
are expressed only in terms of connected graphs having no
articulation-vertex, i.e. no vertex whose deletion would disconnect
the graph.  What is decisive for our aims is the fact that the
embedding-number onto a hsc lattice of an $l$-edge graph in this class
is a polynomial in $d$ of degree $\lfloor l/2\rfloor$ at most\cite{fisga}. Here
$\lfloor l/2\rfloor$ denotes the integer part of $l/2$. Therefore, in spite of the
fact that the HT expansion coefficients of the
(higher)-susceptibilities at order $l$ are polynomials in $d$ of
degree $l$, the susceptibilities can be simply expressed in terms of
the successive derivatives of the magnetic field with respect to the
magnetization which, at the same expansion order, are polynomials in
$d$ of degree $\lfloor l/2\rfloor$ only.  Thus, one can conclude that 
the exact dependence on $d$ of the HT coefficients of the higher
susceptibilities can actually be determined up to order 20, 
using only a sequence of hsc lattices of dimensionalities $1 \le d \le 10$,
by an interpolation in $d$ of the series coefficients!

Let us finally stress that the elements of the coefficients matrix
$f_{2m,l}$ of the HT and low-field expansion for the free-energy of
the spin-1/2 Ising model can be linearly expressed in terms of the
expansion coefficients of the susceptibilities and therefore they also
are polynomials in $d$ of degree $l$.  This property holds in
particular for the Mayer coefficients $b_s(d)=f_{2s,s}(d) $ of the
dimer gas. More details concerning the graphical expansion procedure
 can be found in Subsect. C of the Appendix.
\section{Derivation of Expansions}
%Considering the partition function leading to the Mayer series, we
%note that in volume $V = N$ the sum is dominated by terms with
%$\frac{p}{2} V$ dimers, this dominance increasing with volume, leading
%to the observation that
%\begin{equation}
%\lambda_d(p)=P(p)-\frac{p}{2}{\rm ln}(z(p)).
%\label{3.5}
%\end{equation}
%This result was noted in Refs.[18] and [19].

As mentioned in the introduction we have a second route
 for deriving $\lambda_d(p)$ and
$\lambda_d$ expansions. The key initial step is the computation of the
quantity $\tilde J_i(d)$ from the quantities $b_i(d)$.  The $\tilde
J_i(d)$ depend on the set of $b_n(d)$ with $n \le i$.  The
computations are given in Ref. [\onlinecite{20}] as follows
\begin{equation}
\nonumber
\tilde J_1=0.
\end{equation}
We first find $\tilde J_r^L$, with $\tilde J_1^L=0$, and from $r=2$
on, inductively defined by
\begin{equation}
\tilde J_r^L=\frac{1}{L}\Big\{ S_r-(exp(L\sum_{i=1}^{r-1}\tilde J_i^Lx^i))|_r\Big\}
\label{3.10}
\end{equation}
where
\begin{equation}
S_r=\sum_{p=0}^r\Big\{ (exp(L\sum_ib_i(\frac{x}{2d})^i))|_p\frac{1}{(r-p)!}
\big (\frac{-1}{2(L-1)} \big )^{r-p} \frac{(L-2p)!}{(L-2r)!}\Big\}.
\label{3.11}
\end{equation}
The symbol $|$ with the subscript $j$ indicates the $j$th coefficient
 in the formal power series in $x$. The $\tilde J_r$ are determined from
 the $\tilde J_r^L$ by taking $L$ to infinity.  We may also
 inductively go from the $\tilde J_i$ to the $b_i$ by the same
 formulae.

 This set of relations was first implicitly used in
Ref.[\onlinecite{14}], but not explicitly written down there. Just as
the $b_i(d)$ are the cluster expansion coefficients of a dimer gas,
the $\tilde J_i(d)$ are the cluster expansion coefficients of a
certain polymer gas\cite{14} and these coefficients of the two gases
are related by the development surrounding eqs. (\ref{3.10}) and
(\ref{3.11}).  This is a clean calculation that requires no hard
proof. The $\tilde J_i(d)$ can be proved \cite{3} to be of the form
\begin{equation}
\tilde J_s(d)=\frac{c_{s,r}}{d^r}+\frac{c_{s,r+1}}{d^{r+1}}+...+\frac{c_{s,s-1}}{d^{s-1}} 
\label{3.12}
\end{equation}
with $r \ge s/2$.

Whereas our first development was basically for each $d$ individually,
we will see as with this last equation that the dependence on $d$ is in
the nitty-gritty of this second development. The present treatment
allows us to get results relating the series for different $d$'s.  As
an example, suppose we know the $\tilde J_i(d)$ for $1 \le d \le 10$,
$i \le 20$.  Then one may derive $\tilde J_i(d)$ for $i \le 20$ and
all $d$! (One has enough information to compute all the $c_{s,r}$ for
$i \le 20$.) The same statement holds for the $b_i(d)$ since one may
go between the set of $b_i(d)$ with $i < n$ and the set of $\tilde J_i(d)$
with $i<n$ as mentioned above.

So far all the results dealt with in this section have been true and
rigorously proven. We now turn to the further development, certainly
true but for which we do not yet have a rigorous proof.  We work for a
given $d$ and take as known the $\tilde J_i(d)$ (which as above could
be calculated from the $b_i(d)$). We then compute $\alpha_i(d)$ by
iterations, from $\alpha_i=0$, of 
\begin{equation}
\alpha_k= \big (\tilde J_kp^k\big ) \frac{1}{(1-2\sum_{i=2} i\alpha_i)^{2k}}\big( 1-2\sum_{i=2} i\alpha_i/p \big)^k. 
\label{3.13}
\end{equation}
Then $\lambda_d(p)$ is given by 
\begin{equation}
\lambda_d(p)=Q_1+Q_2
\label{3.14}
\end{equation}
\begin{equation}
Q_1=\frac{1}{2}(p{\rm ln}(2d)-p{\rm ln}p -2(1-p){\rm ln}(1-p)-p)
\label{3.15}
\end{equation}
\begin{equation}
Q_2=\sum_{i=2} \alpha_i-\sum_{k=2}\frac{1}{k}\Big ( 2\sum_{i=2}i
\alpha_i\Big)^k+
\frac{1}{2} p\sum_{k=2}\frac{1}{k}\Big ( 2\sum_{i=2}i\alpha_i/p\Big)^{k}.
\label{3.16}
\end{equation}
$Q_2$ may be developed as a power series in $p$ 
\begin{equation}
Q_2=\sum_2^{\infty} a_k (d) p^k
\label{3.17}
\end{equation}
where $a_k(d)$ is a polynomial in power of $\frac{1}{d}$ with powers
satisfying $k/2 \le r < k$ (as the powers in Eq. (\ref{3.12})), see
Refs.  [\onlinecite{13}] or [\onlinecite{17}].  So for example if we
know $a_k(d)$ for $k \le 20$ and $d \le 10$, then we can deduce
$a_k(d)$ for  $k \le 20$ and all $d$.
To determine $a_k$ only the values 
of $a_k(d)$,  $d=1,..,\lfloor k/2 \rfloor$ are needed,
 the remaining $10 - \lfloor k/2 \rfloor$ values were used to give consistency
 checks for each  $k < 20$. 
This is a consistency check on both the computation 
 of the $b_i(d)$ and of the theory, since as we mentioned above the 
 development of Eq.(\ref{3.13})-Eq.(\ref{3.17}) has not been yet made
 rigorous.


We can deduce the series for $\lambda_d$, Eq. (2) above, basically by
 setting $p=1$ in Eq.(\ref{3.14}). It is important to note for this
 that each power of $\frac{1}{d}$ gets a contribution from only a
 finite number of $a_k(d)$. Specifically $1/d^s$ get contributions
 from those $a_k(d)$ for which $k/2 \le s < k$. For example if we know
 $a_k(d)$ for $k \le 20$, then we can deduce the terms in $\lambda_d$
 up to $1/d^{10}$.
 
 To get at the theory (of the formal argument leading to
 eqs.(\ref{3.14})-(\ref{3.17}), our second development of the
 $\lambda_d(p)$ and $\lambda_d$ expansions), we recommend to the
 reader starting by reading Ref.[\onlinecite{14}], a three page paper,
 or Section 5 of Ref.[\onlinecite{13}].  We now give a slightly hand
 waving capsule summary of the introductory portion of this theory up
 to the derivation of the mean field formulae Eq.(1) and
 (\ref{lasp1}) above.
 
We work on a periodic $d$-dimensional lattice with number of sites  $N$. A
``difunction'' is a translation invariant periodic function on pairs
of distinct vertices.  We associate to dimers the difunction $f$, that
is $1$ if the pair of vertices are nearest neighbors and $0$
otherwise.  We call a sequence $ X_1, X_2,....$ of $pN$ distinct
vertices a ``$p$-sequence''. We let $\sum$ denote the sum over all 
$p$-sequences. We note that the number of distinct dimer coverings that 
cover a fraction $p$ of the vertices can be represented as 
\begin{equation}
\frac{1}{2^{(pN/2)}} \frac{1}{(pN/2)!} 
\sum \mathop{\prod_{i=1}^{pN}}_{i\text{ odd}} f( X_i, X_{i+1}).
\label{3.18}
\end{equation}
The numerical factors before the sum divide by the number of different
p-sequences that correspond to the same choice of dimers.
The sum is over $\frac{N!}{((1-p)N)!}$ p-sequences.

We let $f_0$ be the difunction of constant value $(\frac {2d}{N-1})$. $f$
and $f_0$ have the same ``normalization'' in the sense that, if one
fixes its first component and sums over the second, one gets the same
answer for both functions.  Replacing $f$ in Eq.(\ref{3.18}) by $f_0$
and using the Stirling formula gives the mean field answer
\begin{equation}
\nonumber
\exp \lambda_{mf}N
\end{equation}
for the number of our dimer covers, where $ \lambda_{mf}$ is as in
Eq. (\ref{lasp1}). 

We write
\begin{equation}
f=f_0+ \cal V
\label{3.19}
\end{equation}
with 
\begin{equation}
{\cal V} =f-f_0.
\label{3.20}
\end{equation}
Expansions in powers of $ \cal V$ may be converted into the expansions 
of this paper.
\section{Numerical estimates}
It is interesting at this point to get some feeling about the accuracy
of the estimates of $h_d$ and $\tilde h_d \equiv \lambda_d(1)$ that
can be obtained from the expression Eq.(\ref{laspd1}) for
$\lambda_d(p)$ when a sufficiently large number of coefficients
$a_k(d)$ is known.  For the evaluation of both $h_d$ and $\tilde h_d $
a first orientation comes from truncating the expansion
$\sum_{k=2}^{\infty}a_k(d)p^k$ at the order $k=r$ and plotting the result
vs. some power of $1/r$.  Let us first consider the quantity $\tilde
h_d$. Assume that the series converges for $p=1$ and that the
coefficients $a_k(d)$ are all positive, then its successive
truncations must provide an increasing sequence of lower bounds of the
limit. Actually, we can check that the approximation of truncating the
expansion at the highest known order is always consistent with the
known upper bounds. However in the case of $d=2$ and $d=3$ the values
thus obtained, i.e. $\tilde h_2=0.2865..$ and $\tilde h_3=0.44916..$
respectively, appear to be still too small.  Therefore one should
properly extrapolate the sequences $(S_r)=\frac{1} {2}({\rm ln}(2d)-1)
+ \sum_2^{r}a_k(d)$ of the truncated expressions. Of course, the
best way to do this depends on the behavior of the sequences. It is
very encouraging to notice that for all values of $d$, the sequences
are smooth and their behavior is well approximated by the simple Ansatz
$S_r= a + b/r^{\alpha}$ and so one has $a \approx \tilde h_d$. This
procedure is very successful. We observe that  $\alpha$   increases with
$d$ and  ranges from $\alpha \approx 1$ for $d=2$ to $\alpha \approx
2.6$ for $d=6$. In dimensionality $d=2$, this Ansatz gives a good fit
of the last 4 - 10 terms of the sequence and the extrapolated value
$a=0.2915(20)$ agrees with the exactly known value $\tilde
h_2=0.291560...$ in Eq.(\ref{lad3}), within the estimated error. The
uncertainty we have written is very conservative, although somewhat
arbitrary. It is obtained allowing for the spread of values resulting
from small variations of the exponent $\alpha$ in the functional form
used for fitting or from a comparison with other extrapolations
obtained for example evaluating the series $\sum_{k=2}^r a_k(d)p^k$
for $p=1$, by Pad\'e or differential approximants\cite{gutt} and
adding the result to the expression $\frac{1} {2}({\rm ln}(2d)-1)$.
Analogously, for $d=3$ the sequence $(S_r)$ is well fitted by the
Ansatz$a+b/n^{1.3}$ and leads to the estimate $\tilde
h_3=0.4499(2)$. This value is not far from the estimate $\tilde
h_3=0.4479$ obtained by a MonteCarlo calculation\cite{beichl} or from
$\tilde h_3=0.453(1)$ obtained\cite{21} extrapolating a much shorter
expansion, and it is also completely consistent with the
bounds\cite{1,6,schrij,lund} $0.440075842 \leq \tilde h_3 \leq 0.4575469308$.

Proceeding along the same lines, we
 can determine the values of $\tilde h_d$ for any value of $d$. We
 notice that the apparent precision of the results improves rapidly
 as $d$ grows, while the differences between the extrapolated values
 and the highest order truncations of the series (as well as the estimated
 uncertainties)  decrease rapidly. The final estimates are always completely 
 consistent with the known bounds. Our estimates of $\lambda_d(1)$
 for $d=2,...,8$ are reported in table \ref{tab3}. Notice that evaluations
of these quantities appear very rarely\cite{finch} in the literature.

The computation of $h_d$ requires only a quite short comment.
Unsurprisingly, the sequences of truncated expansions $\sum_{k=2}^r
a_k(d)p^k$ evaluated for $p< 1$ show a faster convergence than for
$p=1$.  The estimates of $h_d$ thus obtained agree well, within their
uncertainties, with those already listed in Table VII of
Ref.[\onlinecite{bp1}] which have been obtained resumming via Pad\'e
approximants the expansion of $P(z)$ for $z=1$. Therefore the reader
is referred to this source.
\begin{table}[ht]
  \caption{ Our estimates of $\tilde h_d=\lambda_d(1)$ for the 
    (hyper)-simple-cubic lattices of dimensions $d=2,3,...8$ with the 
 rigorous     lower and upper bounds\cite{13} defined by 
 $(1/2){\rm ln}(2d)-1/2 \leq \lambda_d(1) \leq {\rm ln}((2d)!)/4d  $. 
    While these rigorous bounds are valid
    for all $d$, for $d=2$ we have simply reported the known exact value and
  for $d=3$ we have reported the   
    tighter bounds from Refs.[\onlinecite{1,6,schrij,lund}].  
    The non-rigorous lower bounds are simply
    obtained assuming the validity of the positivity conjecture for the 
    coefficients $a_k(d)$ and  truncating  our expansions at the highest 
    available order.}
\begin{tabular}{|c|c|c|c|c|}
 \hline
 & Lower Bound& Non-rig. L.B.& Our Estimate& Upper Bound\\
 \hline
$\lambda_2(1)$&0.29156090&0.286521& 0.2915(20) &0.291560905\\
$\lambda_3(1)$&0.44007584&0.449164& 0.4499(2)  &0.457546931\\
$\lambda_4(1)$&0.53972077&0.576517& 0.57666(3) &0.662787682\\
$\lambda_5(1)$&0.65129254&0.679434& 0.67949(2) &0.755220629\\
$\lambda_6(1)$&0.74245332&0.765301& 0.765315(2)&0.832800604\\
$\lambda_7(1)$&0.81952866&0.838785& 0.838789(1)&0.899686471\\
$\lambda_8(1)$&0.88629436&0.902947& 0.902949(1)&0.958495629\\
 \hline
\end{tabular}
\label{tab3}
\end{table}
\subsection{Series expansion for $h_d$ for $d$ large}
As $d$ goes to infinity, $h_d$ tends to $\tilde{h}_d$.
One can compute the rate with which the former approaches the latter
performing an expansion in $\frac{1}{\sqrt{d}}$.

To compute $h_d$ one looks for a stationary point of Eq.(\ref{laspd1}).
Putting $y = 1/\sqrt{2d}$, the stationarity equation can be written as
\begin{equation}
    (1 - p_{st})^2 - p_{st}y^2 \exp(-2\sum_{k=2}ka_kp_{st}^{k-1}) = 0
\end{equation}
This equation can be solved for large $d$.
Knowing $a_k$ up to $k=20$, one can solve iteratively the equation
up to the order $y^{42}$. Here we shall report only the first few terms
\begin{equation}
p_{st} = 1 - y + \frac{1}{2}y^2 + \frac{3}{8}y^3 - y^4 + \frac{201}{128}y^5 
- \frac{ 5}{2}y^6 + \frac{ 7003}{1024}y^7 - 22y^8 + ...
\label{py}
\end{equation}
At the second order in $y$, it agrees with the value of $p_{st}$ associated to
the lower bound for $h_d$ found in \cite{6,7}
\begin{equation}
  p_{st} = \frac{4d + 1 - \sqrt{8d + 1}}{4d}
\end{equation}
Substituting  Eq.(\ref{py}) into Eq.(\ref{laspd1}), one gets
\begin{equation}
    h_d - \tilde{h}_d = y - \frac{1}{4}y^2 - \frac{11}{24}y^3 + ...
\label{diffh}
\end{equation}
\begin{equation}
    h_d = \frac{1}{2}(\ln{2d} - 1) + \frac{1}{\sqrt{2d}} - \frac{11}{48\sqrt{2d^3}} + O(d^{-2})
\label{hdas}
\end{equation}
in which we wrote only the first three terms out of the $40$ terms we computed.
Using $40$ terms, this series expansion agrees with the difference
$h_d - \tilde{h}_d$ computed numerically
up to $2\cdot10^{-6}$ for $d=7$, $10^{-15}$ for $d=20$.
From $d=40$ up to $d=9000$ the precision is only $10^{-16}$.
Probably the reason for this is that as $d$ increases the point of maximum
$p_{st}$ moves closer to $1$, decreasing the precision.

The terms given in Eq.(\ref{hdas}) give $h_d$ with error less than 
$3\cdot 10^{-3}$ for $7 \leq d < 100$ and 
$2\cdot10^{-5}$ for $100 \leq d < 10000$.

In particular from Eq.(\ref{py}) and Eq.(\ref{diffh}) one gets
\begin{equation}
    \lim_{d \to \infty} \frac{h_d - \tilde{h}_d}{p_{st}(d) - 1} = -1
\end{equation}

\section{Appendix}
\subsection{The conjecture that the coefficients $a_k$ are positive
 in the case of the hsc lattices}
We proved that the coefficients $a_k(d)$ are positive integers for $k
\leq 20$ and $d \geq 1$ by locating in the complex $d$-plane
 their real roots, and counting the
complex ones to make sure that none is missing.  It is interesting to
note that for $1 < d < 2$ or for $2 < d < 3$ the $a_k(d)$ can be
negative, and that there are roots approaching $1$ and $2$ as $k$ gets large.
\begin{table}[ht]
\caption{ Real roots of $a_k(d)$ for $k \geq 10$}
\begin{tabular}{|c|c|c|c|}
 \hline
 $k=10$&  -0.65502486055142554 & 0.99997855862379883 & \\
 $k=11$&  1.0000010707811947 & 1.6603775954637132 & \\
 $k=12$&  -0.12473442164710268 & 1.0000008060184913 & 1.5835444714309055 \\
 $k=13$&  0.99999998817575145 & & \\
 $k=14$&  1.0000000363954472 &1.628126162558255 & \\
 $k=15$&  1.000000000099149 &1.9594209128425236 & \\
 $k=16$&  0.99999999993591589 & & \\
 $k=17$&  0.99999999999934955 & 2.0071302031011769 & 2.4952449117198663 \\
 $k=18$&  1.0000000000002549 & 1.9889667170409254 & \\
 $k=19$&  1.0000000000000033  &1.9993853767904753 & \\
 $k=20$&  0.99999999999999901 & 2.0009898597900763&2.6230186617839066  \\
 \hline
\end{tabular}
\label{tabr}
\end{table}
As we have already noticed, based on the conjecture that the $a_k$ are positive,
the computed values of $a_k$ provide a lower bound of $\lambda_d$.
For $d=2$  in the case of $h_d$, this lower bound  $0.662798966$,
 is smaller than the estimate $0.662798972(1)$ obtained in
Ref.[\onlinecite{bp1}] by Pad\'e approximants. For $d \geq 3$ these
lower bounds  reproduce within the error the Pad\'e
estimates of Ref.[\onlinecite{bp1}].
\subsection{Generalization of the positivity conjecture to other
 bipartite lattices}
There is some evidence that the positivity conjecture can be extended
to other bipartite lattices.  Let us recall what is known on other
lattices.  $\lambda_d$ has been computed from the Mayer coefficients
$b_n$ on other lattices using the formula
\begin{equation}
\lambda_d = -\frac {1} {2} p {\rm ln}(\frac {p} {q}) - (1-p){\rm ln}(1-p)
 - \frac{p}{2} + \frac{q}{2} \sum_{k=2} 
\frac{C_k(\frac{p}{q})^k} {k(k-1)}
\label{other}
\end{equation}
with $q$  the lattice coordination number. The notation $a_k = \frac{q}{2}\frac
{C_k/q^k} {k(k-1)}$ extends that used for the hypercubic
case.

Let us now  report the available data for other bipartite lattices.

In the case of the tetrahedral lattice ($q=4$), taking the Mayer coefficients 
$b_n$ from
Ref.[\onlinecite{kurfis,kenzie}] 
we obtain the following set of coefficients $C_k$

$1,1,1,1,31,253,1261,4897,16201,49501,161239,643969,3006823,14104861,60942421,$
$237903169,854124745,2955594097$

For the bcc lattice, the $b_n$ for $n \leq 24$ have been computed in
Ref.[\onlinecite{bp1}] for $d=3,4,5,6,7$. The coefficients $C_k$
computed from them are all positive. In Table \ref{bcc} 
we list the coefficients $C_k(d)$ for hbcc lattices of dimensionalities  $d=3,4,5$.
The coordination-number is $q=2^d$.
\begin{table}[ht]
\caption{The coefficients $C_k(d)$ in Eq.(\ref{other}) with $ k=2,...,24$ for the 
(hyper)-body-centered-cubic lattices of dimensionalities  $d=3,4,5$.}
\begin{tabular}{|c|c|c|c|}
\hline
&$d=3$&$d=4$&$d=5$\\
\hline
$C_2$&1 & 1 & 1\\
$C_3$&1 & 1 & 1\\
$C_4$&37 & 151 &541\\
$C_5$&241 & 1001 & 3601\\
$C_6$&1651 & 21241 & 290851\\
$C_7$&13861 & 276445 & 4136581\\
$C_8$&109873 & 4138275 & 185447641\\
$C_9$&850465 & 61222177 & 3766174561\\
$C_{10}$&6620401 & 903139171 & 134478272521\\
$C_{11}$&51657541 & 13527055301 & 3251891481301\\
$C_{12}$&403327651 & 201952069177 & 105463232417731\\
$C_{13}$&3151118881 & 3041256137921 & 2794164743354401\\
$C_{14}$&24647038963 & 45839858214697 & 86840903677417891\\
$C_{15}$&192950685061 & 69396577375846 & 2421252466929163141\\
$C_{16}$&1510882839217 & 10530703348244851 & 73870429278903327001\\
$C_{17}$&11833222518145 & 160247978490447425 & 2123026721471921771521\\
$C_{18}$&92728596423613 & 2444106838568935375 & 64306694719829414761621\\
$C_{19}$&727194198560401 & 37359234126615235321 & 1883895461127373802533921\\
$C_{20}$&5707071682914097 & 572176086489368008851 & 56961277210166888567226841\\
$C_{21}$&44820667959807601 & 8779078842662089743601 & 1690242630478526669835704401\\
$C_{22}$&352227866459521537 & 134925544759538198882283 & 51146624643545703193238849401\\
$C_{23}$&2769671081569110445 & 2076868645293925124133493 & 1531526780518608097927545101821\\
$C_{24}$&21790699297032926587 & 32014374542692855556562921 & 46435767644223061358549293433371\\
 \hline
 \end{tabular}
 \label{bcc}
 \end{table}

For the hexagonal lattice \cite{Ftri}, the coefficients $C_k$ are

$1, 1, 1, 1, 11, 85$

Let us now turn to the case of non-bipartite lattices.

For the triangular lattice ($q=6$) the coefficients up to $C_6$ are listed
in Ref.[\onlinecite{Ftri}], while higher-order ones are obtained from
Ref.[\onlinecite{kurfis}]

$1,-3, -11, 1, 91, 141,-1651,-16143,-87329,-295063,-72533,8092033,76819835$

For the fcc lattice ($q=12$), from Ref.[\onlinecite{kurfis}] we obtain:

$1,-7,19,41,-779,3557,46327,118529,-557909$

These data imply that the positivity conjecture for  the $C_k$
 has to be restricted to bipartite lattices.

 On a Bethe lattice\cite{nagle,stilck} the entropy is given by Eq.(\ref{other})
 with $C_k = 1$ for all $k$.
Notice that on any lattice $C_k = 1$ for $k < r$, where $r$ is the
length of the smallest nontrivial loop on the lattice,
because the diagrams contributing to such $C_k$ can't
tell the difference between the given lattice and a
Bethe lattice of the same coordination number.

A stronger form of the positivity conjecture is that $C_k \geq 1$ for
bipartite lattices.
\subsection{Graphical expansion procedure for the Ising  model}
For simplicity, the whole graphical expansion procedure can be split into three
steps.  First, one has to list all graphs entering into the
calculation up to the maximum order $L_{max}$ of expansion. To begin
with, one forms the simple, topologically distinct
one-vertex-irreducible graphs with $l \le L_{max}$ edges.  One can
further restrict to the subset of the bipartite graphs, since only
these can be embedded onto the bipartite hsc or hbcc lattices.
  This is the only memory intensive part of the
procedure, because there are many graphs\cite{bp2,bp3,bp4} 
(approximately $3\cdot 10^5$
graphs at order 20,  and over $5\cdot10^7$ at order 24, 
but it took only a few hours.  In a second
step, the lattice embedding-numbers and the symmetry-numbers of these
graphs are computed, one vertex of these graphs is ``marked'' in all
possible ways and the graphs are "decorated" to have also multiple
lines.  This is the subset of the graphs from which the expansion of
the magnetization can be reconstructed.

In the case of hsc lattices of high dimensionality, the most
time-consuming part of this procedure is the computation of the
embedding-number for each graph. In the case of the hbcc lattices the
timings are much smaller and very slowly dependent on $d$, but
unfortunately the expansion coefficients are not polynomials in $d$.
One begins by appropriately ordering the graph vertices, and then the
first of them is placed at the lattice origin. The possible positions
of the second vertex can be counted exploiting the symmetries of the
hypercube.  After fixing the first two points of the embedding, the
possible positions of the remaining vertices are restricted to
relatively few configurations by the constraints given by the
distances from the first two points and the count can go on in a
relatively easy way.  On the hsc lattices, the timings for computing
the magnetization expansion of the $d$-dimensional Ising model at
order $L_{max}$ increase exponentially with the order of expansion and
the lattice dimensionality $d$: roughly as $O(5.5^{L_{max}} 2.5^d)$.  In
particular, the computation for the $10$-dimensional Ising model at
order $20$ took $42$ days of single-core time on a quad-core desktop
computer with a CPU-clock frequency of $2.8GHz$. Actually, less time
was used since the calculation was appropriately distributed on the
four cores of the computer.  Using more extensive computer resources,
it would be possible to compute only a few more orders, for not too
high lattice dimensionalities.

The next step implements the algebraic ``vertex-renormalization'',
namely the procedure of reconstruction\cite{wortis} of the
magnetization from the one-vertex-irreducible graphs having a single
marked vertex. By integrating the magnetization exactly with respect
to the field one finally obtains the free-energy in terms of the bare
vertices (up to a standard constant of integration).  This step of the
calculation is based on codes written in the Python language and is
fast.  The free-energy thus computed is model independent: eventually
one has to specialize the precise form of the bare vertex-functions to
the particular model of interest.



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\end{document}
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