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Polynomial chaos and scaling limits of disordered systems
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In this paper, the authors formulate general conditions ensuring that a sequence of multi-linear polynomials of independent random variables converges to a limiting random variable, given by a Wiener chaos expansion over the d-dimensional white noise.Abstract:
Inspired by recent work of Alberts, Khanin and Quastel, we formulate general conditions ensuring that a sequence of multi-linear polynomials of independent random variables (called polynomial chaos expansions) converges to a limiting random variable, given by a Wiener chaos expansion over the d-dimensional white noise. A key ingredient in our approach is a Lindeberg principle for polynomial chaos expansions, which extends earlier work of Mossel, O'Donnell and Oleszkiewicz. These results provide a unified framework to study the continuum and weak disorder scaling limits of statistical mechanics systems that are disorder relevant, including the disordered pinning model, the (long-range) directed polymer model in dimension 1+1, and the two-dimensional random field Ising model. This gives a new perspective in the study of disorder relevance, and leads to interesting new continuum models that warrant further studies.read more
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Universality in marginally relevant disordered systems
TL;DR: In this article, the authors consider disordered systems of directed polymer type, for which disorder is so-called marginally relevant, and show that in a suitable weak disorder and continuum limit, the partition functions of these different models converge to a universal limit.
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McKean-Vlasov optimal control: the dynamic programming principle
TL;DR: In this article, the authors study the McKean-Vlasov optimal control problem with common noise in various formulations, namely the strong and weak formulation, as well as the Markovian and non-Markovian formulations, and allow for the law of the control process to appear in the state dynamics.
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The two-dimensional KPZ equation in the entire subcritical regime
TL;DR: In this article, the authors consider the KPZ equation in space dimension 2 driven by space-time white noise and show that the solution admits subsequential scaling limits for sufficiently small values of β.
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Paracontrolled distributions on Bravais lattices and weak universality of the 2d parabolic Anderson model
Jörg Martin,Nicolas Perkowski +1 more
TL;DR: In this article, a version discrete de la theory des distributions paracontrolees comme outil is proposed for deduire les limites d'echelles des modeles discrets.
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On the Moments of the (2+1)-Dimensional Directed Polymer and Stochastic Heat Equation in the Critical Window
TL;DR: In this article, the authors compute the limiting third moment of the space-averaged partition function, showing that it is uniformly bounded, which implies that rescaled partition functions, viewed as a generalized random field on the directed polymer model, have non-trivial subsequential limits and each such limit has the same explicit covariance structure.
References
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Entropy, large deviations, and statistical mechanics
TL;DR: In this paper, the authors introduce the concept of large deviations for random variables with a finite state space, which is a generalization of the notion of large deviation for random vectors.
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Effect of random defects on the critical behaviour of Ising models
TL;DR: In this article, a cumulant expansion is used to calculate the transition temperature of simple-square Ising models with random-bond defects, and the results are -Tc-1 dTc/dx mod x=0.329 compared with the mean-field value of one.
Proceedings ArticleDOI
The influence of variables on Boolean functions
TL;DR: Methods from harmonic analysis are used to prove some general theorems on Boolean functions and enable them to prove theorem on the rapid mixing of the random walk on the cube and in the extremal theory of finite sets.