The Eigenfunctions of the Transfer Operator for the Dyson Model in a Field

Abstract

The recent works [5] and [15] have studied the spectral properties of the Dyson model in the absence of an external field. This paper is a continuation of [5] 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 zeromagnetic- field case, this eigenfunction is not continuous.