Localization of the Gaussian multiplicative chaos in the Wiener space and the stochastic heat equation in strong disorder

TL;DR

This paper investigates the localization phenomena of Gaussian multiplicative chaos on Wiener space, revealing a transition to a glassy phase where endpoint distributions become purely atomic in the low temperature regime.

Contribution

It demonstrates the freezing of the energy landscape and the emergence of a glassy phase with localized endpoint distributions for GMC in Wiener space, extending localization results to this setting.

Findings

Endpoint distribution localizes in few spatial islands at low temperature.

Total mass of GMC becomes purely atomic in the glassy phase.

System enters a frozen, glassy phase with localized endpoint distribution.

Abstract

We consider a {\it{Gaussian multiplicative chaos}} (GMC) measure on the classical Wiener space driven by a smoothened (Gaussian) space-time white noise. For $d\geq 3$ it was shown in \cite{MSZ16} that for small noise intensity, the total mass of the GMC converges to a strictly positive random variable, while larger disorder strength (i.e., low temperature) forces the total mass to lose uniform integrability, eventually producing a vanishing limit. Inspired by strong localization phenomena for log-correlated Gaussian fields and Gaussian multiplicative chaos in the finite dimensional Euclidean spaces (\cite{MRV16,BL18}), and related results for discrete directed polymers (\cite{V07,BC16}), we study the endpoint distribution of a Brownian path under the {\it{renormalized}} GMC measure in this setting. We show that in the low temperature regime, the energy landscape of the system freezes and enters the so called {\it{glassy phase}} as the entire mass of the Ces\`aro average of the endpoint GMC distribution stays localized in few spatial islands, forcing the endpoint GMC to be {\it{asymptotically purely atomic}} (\cite{V07}). The method of our proof is based on the translation-invariant compactification introduced in \cite{MV14} and a fixed point approach related to the cavity method from spin glasses recently used in \cite{BC16} in the context of the directed polymer model in the lattice.