Lace expansion for the Ising model

TL;DR

This paper extends the lace expansion technique to the Ising model for all coupling strengths, establishing bounds and Gaussian asymptotics for high-dimensional cases without relying on reflection positivity.

Contribution

It proves the lace expansion for the Ising model across all coupling regimes and derives Gaussian asymptotics in high dimensions without reflection positivity assumptions.

Findings

Lace expansion valid for any spin-spin coupling in the Ising model.

Diagrammatic bounds similar to those in self-avoiding walk are established.

Gaussian asymptotics for the critical two-point function in high dimensions.

Abstract

The lace expansion has been a powerful tool for investigating mean-field behavior for various stochastic-geometrical models, such as self-avoiding walk and percolation, above their respective upper-critical dimension. In this paper, we prove the lace expansion for the Ising model that is valid for any spin-spin coupling. For the ferromagnetic case, we also prove that the expansion coefficients obey certain diagrammatic bounds that are similar to the diagrammatic bounds on the lace-expansion coefficients for self-avoiding walk. As a result, we obtain Gaussian asymptotics of the critical two-point function for the nearest-neighbor model with d>>4 and for the spread-out model with d>4 and L>>1, without assuming reflection positivity.