Strong Disorder for Stochastic Heat Flow and 2D Directed Polymers

TL;DR

This paper studies the 2D stochastic heat flow and directed polymers under strong disorder, revealing sharp extinction rates, phase transition scales, and fluctuation behaviors in various regimes.

Contribution

It provides a unified framework for analyzing the large-time behavior and phase transitions of 2D stochastic heat flow and directed polymers under strong disorder.

Findings

Identifies the rate of local extinction in the 2D stochastic heat flow.

Determines the spatial scale for transition from vanishing to diverging mass.

Establishes free-energy estimates for 2D directed polymers.

Abstract

The critical 2D Stochastic Heat Flow (SHF) is a universal measure-valued process that provides a notion of solution to the ill-defined 2D stochastic heat equation. We investigate the SHF in the large-time and strong-disorder regimes, proving a sharp form of local extinction: we identify the rate at which the distribution collapses to zero. We also identify the spatial scale governing the transition from vanishing mass to diverging mass, and from extinction to an averaged behavior. Corresponding results are established for the partition functions of 2D directed polymers, yielding precise free-energy estimates. Our proof provides a unified framework of change of measure and coarse-graining arguments. These results offer new insights into the 2D stochastic heat equation regularized via space-time discretization: for any regime of supercritical disorder strength $\beta$, including the case where $\beta > 0$ is kept fixed, the solution exhibits fluctuations on a superdiffusive scale.