Continuous eigenfunctions of the transfer operator for Dyson models

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 $\text{J}(k)=\beta k^{-\alpha }$, $k\ge 1$, in the whole subcritical regime $\beta <{\beta }_c$ for which the parameter $\alpha$ is greater than $3\mathrm{/}2$.