Purdue Probability Seminar, Spring 2023

Random walks in cooling random environments (RWCRE) are a model of random walks in dynamic random environments where the random environment is re-sampled at a fixed sequence of times (called the cooling sequence) and the environment remains constant between these re-sampling times. In this talk we will consider the limiting distribution of the RWCRE when the underlying distribution of the environment is such that the random walk in a fixed environment is in the "borderline CLT" regime; that is, the walk in a fixed environment converges to a Gaussian but with scaling \(\sqrt{n\log n}\) rather than diffusive \(\sqrt{n}\) scaling. We show that for any choice of a cooling sequence the RWCRE also converges in distribution to a Gaussian, but the proper normalization is surprisingly delicate.

We will show how probability distributions of first exit times of \(\alpha\)-stable symmetric processes from a bounded domain \(D\) increase through the use of Steiner symmetrization. We will also show how, when a sequence of Lipschitz domains \(\{D_n\}\) converges to a domain \(D'\) with respect to the Hausdorff metric, the corresponding sequence of probability distributions of the first exit time of Brownian motion from \(D_n\) converges to the distribution of the first exit time of Brownian motion from \(D'\). These results will be then applied to establish results on distributions of first exit times on specific sequences of domains such as triangles and quadrilaterals.

This paper develops the large deviations theory for the point process associated with the Euclidean volume of k-nearest neighbor balls centered around the points of a homogeneous Poisson or a binomial point processes in the unit cube. Two different types of large deviation behaviors of such point processes are investigated. Our first result is the Donsker-Varadhan large deviation principle, under the assumption that the centering terms for the volume of k-nearest neighbor balls grow to infinity more slowly than those needed for Poisson convergence. Additionally, we also study large deviations based on the notion of -topology, which takes place when the centering terms tend to infinity sufficiently fast, compared to those for Poisson convergence. As applications of our main theorems, we discuss large deviations for the number of Poisson or binomial points of degree at most k in a geometric graph in the dense regime.

his is joint work with Christian Hirsch (Aarhus) and Takashi Owada (Purdue).

In a disordered system one can ask whether random fluctuations overcome the tendency to localize around energy minimizing ground states. In this talk I will discuss localization in the directed polymer model and introduce the concept of joint localization. Joint localization occurs when polymer measures with different initial conditions all localize in a common region of space. Under certain non-degeneracy and monotonicity conditions of the polymer measures we show that single polymer localization implies joint localization. We verify these general conditions in two specific cases: directed polymers on the lattice and Gaussian directed polymers. This talk is based on joint work with Yuri Bakhtin.

Let \(X\) be a \(d\)-dimensional Gaussian process on \(\mathbb{R}_+\), where the component are independents copies of a scalar Gaussian process \(X_0\) on \(\mathbb{R}_+\) with a given general variance function \(\gamma^2(r)=\operatorname{Var}\left(X_0(r)\right)\) and a canonical metric \(\delta(t,s):=(\mathbb{E}\left(X_0(t)-X_0(s)\right)^2)^{1/2}\) which is commensurate with \(\gamma(|t-s|)\). We provide some general conditions on \(\gamma\) so that for any Borel set \(E\subset [0,1]\), the Hausdorff dimension of the image \(X(E)\) and the graph \(Gr_E(X)\) are constants almost surely.

We construct and study properties of an infinite dimensional analog of Kahane's theory of Gaussian multiplicative chaos. Namely, we consider a random field defined with respect to space-time white noise integrated w.r.t. Brownian paths in \(d\geq 3\) and construct the infinite volume limit of the normalized exponential of this field, weighted w.r.t. the Wiener measure, in the entire weak disorder (subcritical) regime. Moreover, we characterize this infinite volume measure, which we call the subcritical GMC on the Wiener space, w.r.t. the mollification scheme in the sense of Shamov and determines its support by identifying its thick paths. This, in turn, implies that, almost surely, the subcritical GMC on the Wiener space is singular w.r.t. the Wiener measure. We also prove, in the subcritical regime, the existence of negative and positive (\(L^p\) for \(p>1\)) moments of the total mass of the limiting GMC and deduce its Hölder exponents (small ball probabilities) explicitly. While the uniform Hölder exponent (the upper bound) and the pointwise scaling exponent (the lower bound) differ for a fixed disorder, we show that, as the disorder goes to zero, the two exponents agree, coinciding with the scaling exponent of the Wiener measure.

This is joint work with Isabel Lammers and Chiranjib Mukherjee.

This talk presents the convergence of the KPZ equation to the directed landscape, which is the central object in the KPZ universality class. This convergence result is the first to the directed landscape among the positive temperature models.

Stochastic homogenization studies the convergence of PDE with random coefficients to a deterministic "effective" equation. A closely related problem is the quenched central limit theorem (QCLT) for random walks in random environment (RWRE) which states that the RWRE converges to a Brownian motion with deterministic diffusivity in large scale. In this talk we consider the stochastic homogenization of non-divergence form equations on the integer lattice and the corresponding RWRE (which is a martingale). We will derive the optimal rates of the homogenization. We will also discuss the correlation structure of the invariant measure and quantitative estimates for the QCLT. Joint work with Hung V. Tran (UW-Madison).

In this talk, I will present several results on the Anderson Hamiltonian with white noise potential in dimension 1. This operator formally writes « - Laplacian + white noise ». It arises as the scaling limit of various discrete models and its explicit potential allows for a detailed description of its spectrum. In particular, we will discuss various techniques to prove Anderson localization for the model on the full line. Joint works with Cyril Labbé.

In this talk, I will first give an overview of stochastic homogenization for nondivergence form elliptic equations (from the PDE perspective) and quenched invariance principles for nonreversible diffusion processes (from the probability perspective). I will then present new quantitative homogenization results for the parabolic Green Function (fundamental solution) and for the unique ergodic invariant measure. This invariant measure is a solution of the adjoint equation in doubly divergence form, satisfying certain integrability conditions. I will discuss the implications of these homogenization results, such as heat kernel bounds on the parabolic Green function and quantitative ergodicity for the environmental process. This talk is based on joint work with Scott Armstrong (NYU) and Benjamin Fehrman (Oxford).

In this talk, I will discuss long-time dynamical behaviors of Langevin dynamics, including Langevin dynamics on Lie groups and mean-field underdamped Langevin dynamics. We provide unified Hessian matrix conditions for different drift and diffusion coefficients. This matrix condition is derived from the dissipation of a selected Lyapunov functional, namely the auxiliary Fisher information functional. We verify the proposed matrix conditions in various examples. I will also talk about the application in distribution sampling and optimization. This talk is based on several joint works with Erhan Bayraktar and Wuchen Li.