Semi-classical analysis of non self-adjoint transfer matrices in statistical mechanics. I

TL;DR

This paper introduces a semi-classical approach to analyze transfer operators with complex actions in one-dimensional statistical mechanics, focusing on estimating powers of approximately normal integral operators.

Contribution

It develops a semi-classical analysis framework for non self-adjoint transfer matrices, advancing the understanding of their spectral properties in complex-valued models.

Findings

Semi-classical estimates for powers of integral operators.

Analysis of transfer operators near normality.

Application to complex-valued action models.

Abstract

We propose a way to study one-dimensional statistical mechanics models with complex-valued action using transfer operators. The argument consists of two steps. First, the contour of integration is deformed so that the associated transfer operator is a perturbation of a normal one. Then the transfer operator is studied using methods of semi-classical analysis. In this paper we concentrate on the second step, the main technical result being a semi-classical estimate for powers of an integral operator which is approximately normal.