Investigation of the Conditions of the Asymptotic Existence of the Configuration Integral of the Gibbs Distribution

R. L. Dobrušin

Teoriâ veroâtnostej i ee primeneniâ, Tome 9 (1964) no. 4, pp. 626-643

Résumé

Let $V$ be a cube of dimension $\nu$, with volume $|V|$. Let ${{|V|}/{N \to\nu}}$, $N\to\infty$. Let ${\mathbf x}=(x_1,\cdots ,x_N)$, $x_i\in V$, $i=1,\cdots ,N$, $$ Q(V,N)=\int_V\dotsi\int_V\exp\{-\beta U({\mathbf x})\}\,dx_1\dots dx_N, $$ where $$ U({\mathbf x})=\sum_{1\leqq i<j\leqq N}\Phi(|x_i-x_j|). $$ The conditions on $\Phi(y)$, which are sufficient and in some sense necessary for the existence of the finite limit $$ \lim_{N\to\infty}\frac1N\log\frac1{{N!}}Q(V,N) $$ are given.