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

TL;DR

This work analyzes a directed polymer in a random environment on the infinite supercritical percolation cluster in $\mathbb{Z}^d$ ($d\ge 3$). By combining ergodicity, a block/tube geometric construction, and concentration and second-moment methods, the authors establish that $\beta_c(\mathcal{C}_\infty)=0$ while $\overline{\beta}_c(\mathcal{C}_\infty)>0$, revealing a non-empty strong-disorder subphase where very strong disorder fails. They further show a deterministic lower bound $\overline{\beta}_c^{\mathrm{cluster}}\ge\beta_{L^2}(\mathbb{Z}^d)>0$ and prove almost-exponential decay rates for $W_{n,\mu}^\beta$ in this subphase, highlighting a qualitative difference from the sharp phase transition on the full lattice. The results underscore how random geometric structures like percolation clusters influence disorder relevance and phase structure in directed polymers, with precise mechanisms built around good-block/tube geometry and multi-scale probabilistic controls.

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.

Theorems & Definitions (20)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• proof : Proof of Theorem \ref{['thm:MainTheorem']}, part (i)
• Lemma 4.1
• proof
• Lemma 4.2
• proof
• Lemma 5.1