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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