Continuous eigenfunctions of the transfer operator for Dyson models

Abstract

In this article we address a well known problem at the intersection of ergodic theory and statistical mechanics. We prove that there exists a continuous eigenfunction for the transfer operator corresponding to pair potentials that satisfy a square summability condition on the variations, when the inverse temperature is subcritical. As a corollary we obtain a continuous eigenfunction for the classical Dyson model, with interactions $\J(k)=β\, k^{-α}$, $k\ge 1$, in the whole subcritical regime ${\hat{I}}^2<{\hat{I}}_c^2$ for which the parameter $\hat{I}\pm$ is greater than $3/2$.