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

Abstract

We consider a 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 (Electron. Commun. Probab. 21 (2016) 61) 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 (Ann. Appl. Probab. 26 (2016) 643–690; Adv. Math. 330 (2018) 589–687), and related results for discrete directed polymers (Probab. Theory Related Fields 138 (2007) 391–410; Bates and Chatterjee (2016)), we study the endpoint distribution of a Brownian path under the 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 glassy phase as the entire mass of the Cesàro average of the endpoint GMC distribution stays localized in few spatial islands, forcing the endpoint GMC to be asymptotically purely atomic (Probab. Theory Related Fields 138 (2007) 391–410). The method of our proof is based on the translation-invariant compactification introduced in (Ann. Probab. 44 (2016) 3934–3964) and a fixed point approach related to the cavity method from spin glasses recently used in (Bates and Chatterjee (2016)) in the context of the directed polymer model in the lattice.