Random path representation and sharp correlations asymptotics at high-temperatures

TL;DR

This paper introduces a new approach for deriving precise asymptotics of correlation functions in high-temperature statistical mechanics models, extending to complex correlations and connecting to quantum field theory insights.

Contribution

It presents a nonperturbative proof of Ornstein-Zernike asymptotics for 2-point functions and extends results to arbitrary odd-odd correlations in the Ising model.

Findings

Proved sharp asymptotics for correlation functions at high temperatures.

Extended the proof to complex correlation functions in the Ising model.

Linked statistical mechanics results to quantum field theory frameworks.

Abstract

We recently introduced a robust approach to the derivation of sharp asymptotic formula for correlation functions of statistical mechanics models in the high-temperature regime. We describe its application to the nonperturbative proof of Ornstein-Zernike asymptotics of 2-point functions for self-avoiding walks, Bernoulli percolation and ferromagnetic Ising models. We then extend the proof, in the Ising case, to arbitrary odd-odd correlation functions. We discuss the fluctuations of connection paths (invariance principle), and relate the variance of the limiting process to the geometry of the equidecay profiles. Finally, we explain the relation between these results from Statistical Mechanics and their counterparts in Quantum Field Theory.