The Eigenfunctions of the Transfer Operator for the Dyson model in a field

TL;DR

This paper investigates the spectral properties of the Dyson model in a magnetic field, proving the existence of a non-negative eigenfunction of the transfer operator under certain conditions, highlighting differences from the zero-field case.

Contribution

It extends previous spectral analysis of the Dyson model by establishing the existence of a non-negative eigenfunction in the presence of a magnetic field for specific parameter ranges.

Findings

Existence of a non-negative, integrable eigenfunction for high temperatures or strong fields.

Eigenfunction is not continuous in the magnetic field case.

Results hold for ta in (rac{3}{2}, 2].

Abstract

The recent works \cite{EFMV2024} and \cite{JOP2023} have studied the spectral properties of the Dyson model in the absence of an external field. This paper is a continuation of \cite{EFMV2024} and aims to bridge the gap in the literature by investigating the Dyson model in a field.\\ In this paper, we prove that, for high temperatures or strong magnetic fields, there exists a non-negative, integrable (with respect to the unique half-line Gibbs measure) eigenfunction of the transfer operator for the Dyson model if $\alpha\in(\frac 3 2,2]$. However, unlike in the zero-magnetic-field case, this eigenfunction is not continuous.