Non-coincidence of critical points for directed polymers on supercritical percolation clusters

TL;DR

This paper investigates the phase transition behavior of directed polymers on supercritical percolation clusters, revealing a sub-phase within the strong disorder regime where very strong disorder does not occur, contrasting with full lattice results.

Contribution

It demonstrates the existence of a non-empty sub-phase of the strong disorder regime on percolation clusters, showing a nuanced phase transition structure.

Findings

Existence of a sub-phase within the strong disorder regime.

Contrast with sharp phase transition results on full lattices.

Almost sure results for the percolation cluster environment.

Abstract

We consider the model of a directed polymer in a random environment defined on the infinite cluster of supercritical Bernoulli bond percolation in dimensions $d \geq 3$. For this model, it was proved in arXiv:2205.06206 that for almost every realization of the cluster, the polymer is in a strong disorder regime for any positive inverse temperature. Here, we show for almost every realization of the cluster the existence of a non-empty sub-phase of the strong disorder regime, consisting of positive inverse temperatures in which very strong disorder does not hold. This is in contrast to the recently established sharpness of the phase transition for the directed polymer on the full lattice, see arXiv:2402.02562, arXiv:2502.04113.