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In this article we prove that there exists a continuous eigenfunction for thetransfer operator corresponding to potentials for the classical Dyson model inthe subcritical regime for which the parameter $\alpha$ is greater than $3/2$,and we conjecture that this value is sharp. This is a significant improvementon previous results where the existence of a continuous eigenfunction of thetransfer operator was only established for general potentials satisfyingsummable variations, which would correspond to the parameter range $\alpha >2$. Moreover, this complements as result by Bissacot, Endo, van Enter and Le Ny\cite{vanenter}, who showed that there is no continuous eigenfunction at lowtemperatures. Our approach to obtaining these new results involves a novelapproach based on random cluster models.

Continuous eigenfunctions of the transfer operator for the Dyson model