 222 Paul Federbush
 On the PerniciWanless Expansion for the Entropy (and Virial Coefficients) of a Dimer Gas on an Infinite Regular Lattice
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Jan 6, 22

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Abstract. We work with the following expression for the entropy (density) of a
dimer gas on an infinite rregular lattice,
lambda(p) = 1/2 [p ln(r)  p ln(p)  2(1p) ln(1p)  p]
+ Sum_{k=2}^\infty d_k p^k,
where the indicated sum converges for density, p, small enough. Pernici
has computed the coefficients d_k for k \leq 12. He found these d_k to
be polynomials in certain interesting "geometric quantites" arising in
the work of Wanless. Each of these quantities is the number density of
isomorphic mappings of some graph into the lattice (graph). So for a
bipartite lattice,
d_2 = c_2,
d_3 = c_3,
d_4 = c_4 + c_5 \hat{G}_1,
d_5 = c_6 + c_7 \hat{G}_1.
The c_i depend only on r. Here \hat{G}_1 is the density of mapping
classes of the four loop graph into the lattice. The limit of 1/V times
the number of such mapping classes into a lattice of volume V as V goes
to infinity. The infinite volume limit.
There is a simple linear relation that yields the k^{th} virial
coefficient from the value of d_k! We feel this expression gives the
deepest insight into the virial coefficients so far obtained.
What we show in this paper is that such polynomial relations for the d_k
in these geometric quantities holds for the d_k for k \leq 27. Of
course we expect it to hold for all k. We use the same computation
procedure as Pernici. We note this procedure is not rigorously
established. So far a procedure for the physicist, perhaps not the
mathematician (their loss). It is a worthy challenge for the
mathematical physicist to supply the needed rigor.
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