The Aesthetic Asymptotics of the Mayer Series Coefficients for a Dimer Gas on a Regular Lattice
| Autor: | Federbush, Paul |
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| Rok vydání: | 2024 |
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| Druh dokumentu: | Working Paper |
| Popis: | We conjecture that for \textbf{all} regular lattices $b(n)$ is asymptotically of the form in eq.(A1). \begin{equation}\label{A1} \tag{A1} (-1)^{n+1} * b(n) \sim \exp{( k_{-1} n + k_{0} ln(n) + \frac{k_{1}}{n} + \frac{k_{2}}{n^{2}}...)} \end{equation} We restrict testing this to lattices for which we know the first 20 Mayer series coefficients, the $b(n)$. This includes the infinite number of rectangular lattices, one for each dimension, the tetrahedral lattice ( in this one case we know only the first 19 coefficients ), and the (bipartite) body centered cubic lattices, in dimensions 3 through 7. In this paper we will detail results for the rectangular lattices in dimensions 2,3,5,11,and 20, for the tetrahedral lattice, and for the body centered cubic lattices in dimensions 3,4, and 5. These are all bipartite, unfortunately we do not have an example of a non-bipartite regular lattice for which we know enough of the $b(n)$ to work with. For the triangular lattice, regular and non-bipartite, we know the first 14 $b(n)$. We feel this is not enough terms to make any judgement, hopefully someone may compute more terms. We work with an 'approximation' that keeps the first four terms, in $k_{-1}, k_{0}, k_{1}, k_{2}$, in the exponent in eq.(A1). Agreement will be striking. At the end of Part 1 there is a discussion of computations in the present work relevant to conjectures rising from the Yang-Lee Edge Singularity. In Part 7 there is some study of the susceptibility of the Ising model on the 2d rectangular lattice, triangular lattice, and honeycomb lattice; where there is some surprising similarity to the current study of the Mayer series on regular graphs. Comment: 13 pages, replacement of 2d Ising model susceptibility |
| Databáze: | arXiv |
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