Representations and inequalities for Ising model Ursell functions

Abstract

We describe and investigate representations for the Ursell functionun of a family ofn random variables {σi}. The representations involve independent but identically distributed copies of the family. We apply one of these representations in the case that the random variables are spins of a finite ferromagnetic Ising model with quadratic Hamiltonian to show that (−1)n/2+1un(σ1, ...,σn) ≧ 0 forn=2, 4, and 6 by proving the stronger statement\(( - 1 )^{\frac{n}{2} + 1} \frac{{\partial ^m }}{{\partial J_{i1j1} \cdots \partial J_{imjm} }}Z^{\frac{n}{2}} u_n \left| {_{J = 0} } \right. \geqq {}^\backprime 0\) forn=2, 4, and 6, theJij being coupling constants in the Hamiltonian andZ the partition function. For generaln we combine this result with various reductions to show that sufficiently simple derivatives of (−1)n/2+1Zn/2un, evaluated at zero coupling, are nonnegative. In particular, we conclude that (−1)n/2+1un ≧ 0 if all couplings are nonzero and the inverse temperature β is sufficiently small or sufficiently large, though this result is not uniform in the ordern or the system size. In an appendix we give a simple proof of recent inequalities which boundn-spin expectations by sums of products of simpler expectations.