Localization of the Gaussian multiplicative chaos in the Wiener space and the stochastic heat equation in strong disorder
Localization of the Gaussian multiplicative chaos in the Wiener space and the stochastic heat equation in strong disorder
复制标题
维纳空间中高斯乘法混沌的局域化与强无序下的随机热方程
DOI：
10.1214/19-aap1491
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发表时间：
2018
期刊：
影响因子：
--
通讯作者：
Chiranjib Mukherjee
中科院分区：
文献类型：
--
作者：
Yannic Broker;Chiranjib Mukherjee
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.
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.
影响因子：
0.5
作者：
Berestycki, Nathanael
通讯作者：
Berestycki, Nathanael