无限正则格点上二聚气体熵（和维里系数）的Pernici-Wanless展开
On the Pernici-Wanless Expansion for the Entropy ( and Virial Coefficients ) of a Dimer Gas on an Infinite Regular Lattice
摘要：
我们用下面的表达式来计算无限r正则格点上二聚体气体的熵（密度）lambda（p）= 1/2 [ pln（r）-ln（p）-2（1-p）ln（1-p）-p ]+ sum_{k=2}（d_k）（p^k），其中所指示的和在密度p足够小时收敛。Pernici计算了k < 13时的系数d_k。他发现这些d_k是多项式在某些有趣的“几何quantites”所产生的工作万利斯。这些量中的每一个都是某个图到格（图）的同构映射的数密度。所以对于二分格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. c_i只依赖于r。这里的hat{G}_1是四圈图到格的映射类的密度。当V趋于无穷大时，1/V乘以这样的映射类到体积为V的格中的数量的极限。无限容量限制。有一个简单的线性关系，从d_k的值产生第k维里系数！我们认为这个表达式给出了迄今为止得到的维里系数的最深刻的见解。本文证明了，当k < 28时，这些几何量中d_k的多项式关系式对d_k也成立。当然，我们希望它对所有k都成立。我们使用与Pernici相同的计算过程。我们注意到，这一程序并没有严格确立。到目前为止，物理学家的程序，也许不是数学家（他们的损失）。对数学物理学家来说，提供所需的严格性是一个有价值的挑战。
英文摘要：
We work with the following expression for the entropy (density) of a dimer gas on an infinite r-regular lattice lambda(p) = 1/2 [ pln(r)-ln(p)-2(1-p)ln(1-p)-p ]+sum_{k=2}(d_k)(p^k) where the indicated sum converges for density, p, small enough. Pernici has computed the coefficients d_k for k < 13. 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 kth 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 < 28. 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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