Absence of weak disorder for directed polymers on supercritical percolation clusters

TL;DR

This paper proves that directed polymers on supercritical percolation clusters in dimensions three and higher are in a strong disorder phase for almost all clusters and positive temperatures, resolving a previously open question.

Contribution

It establishes the absence of weak disorder for directed polymers on supercritical percolation clusters in high dimensions, a significant advancement in understanding disorder phases.

Findings

Directed polymers exhibit strong disorder on supercritical clusters in $d \,\geq\, 3$.

Almost every cluster realization leads to strong disorder at positive temperatures.

Answers an open question from prior research on disorder phases.

Abstract

We study the directed polymer model on infinite clusters of supercritical Bernoulli percolation containing the origin in dimensions $d \geq 3$, and prove that for almost every realization of the cluster and every strictly positive value of the inverse temperature, the polymer is in a strong disorder phase, answering a question from Cosco, Seroussi, and Zeitouni, see arXiv:2010.09503.