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Localization of the Gaussian multiplicative chaos in the Wiener space and the stochastic heat equation in strong disorder. (English) Zbl 1432.60088
Summary: 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 [the second author et al., Electron. Commun. Probab. 21, Paper No. 61, 12 p. (2016; Zbl 1348.60094)] 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 [T. Madaule et al., Ann. Appl. Probab. 26, No. 2, 643–690 (2016; Zbl 1341.60094); M. Biskup and O. Louidor, Adv. Math. 330, 589–687 (2018; Zbl 1409.60053)], and related results for discrete directed polymers [V. Vargas, Probab. Theory Relat. Fields 138, No. 3–4, 391–410 (2007; Zbl 1113.60097); E. Bates and the second author, “The endpoint distribution of directed polymers”, Preprint, arXiv:1612.03443], 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 [Vargas, loc. cit.]. The method of our proof is based on the translation-invariant compactification introduced in [the second author and S. R. S. Varadhan, Ann. Probab. 44, No. 6, 3934–3964 (2016; Zbl 1364.60037)] and a fixed point approach related to the cavity method from spin glasses recently used in [Bates and the second author, loc. cit.] in the context of the directed polymer model in the lattice.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J65 Brownian motion
60J55 Local time and additive functionals
60F10 Large deviations
35R60 PDEs with randomness, stochastic partial differential equations
35Q82 PDEs in connection with statistical mechanics
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
82D60 Statistical mechanics of polymers
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