Annals of Applied Probability

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

Yannic Bröker and Chiranjib Mukherjee

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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.

Article information

Source
Ann. Appl. Probab., Volume 29, Number 6 (2019), 3745-3785.

Dates
Received: December 2018
First available in Project Euclid: 7 January 2020

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1578366325

Digital Object Identifier
doi:10.1214/19-AAP1491

Mathematical Reviews number (MathSciNet)
MR4047991

Zentralblatt MATH identifier
07172345

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60J65: Brownian motion [See also 58J65] 60J55: Local time and additive functionals 60F10: Large deviations 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15] 35Q82: PDEs in connection with statistical mechanics 60H15: Stochastic partial differential equations [See also 35R60] 82D60: Polymers

Keywords
Gaussian multiplicative chaos supercritical renormalization glassy phase freezing stochastic heat equation strong disorder asymptotic pure atomicity translation-invariant compactification

Citation

Bröker, Yannic; Mukherjee, Chiranjib. Localization of the Gaussian multiplicative chaos in the Wiener space and the stochastic heat equation in strong disorder. Ann. Appl. Probab. 29 (2019), no. 6, 3745--3785. doi:10.1214/19-AAP1491. https://projecteuclid.org/euclid.aoap/1578366325


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