Quenched local CLT for random walks in dynamical balanced random environment
We consider a random walk on Z^d in a balanced random environment which is ergodic with respect to the space-time shift and derive an almost sure local central limit theorem for the corresponding heat kernel. This local CLT follows from a quenched invariance principle and a parabolic Harnack inequality for the corresponding adjoint operator. Joint work with X. Guo
Singularity analysis for heavy-tailed random variables
Heavy-tailed random variables play an important role in applied probability and featuring prominently in insurance mathematics. The joint distribution of $n$ independent identically integer-valued distributed random variables, conditional on the value of the sum, also appears as the stationary distribution for the zero-range process, a continuous-time Markov process for particles distributed on $n$ lattice sites from statistical mechanics. In this context the transition in large deviations from Cramer-type collective behavior ("small steps") to big-jump is interpreted as a transition from gas to condensed phase. The talk presents local large deviations theorems for sums of i.i.d. $\mathbb N_0$-valued, heavy-tailed random variables. The proofs are inspired by singularity analysis and random combinatorial structures. Based on joint work with N. M. Ercolani and D. Ueltschi (J. Theoret. Probab., 2018).
Stable laws for Sinai billiards with cusps at flat points
We consider the dynamical system of Sinai billiards with a cusp where two walls of the billiard table meet at the vertex of a cusp and have zero one-sided curvature, forming a "flat point" at the vertex. For Holder continuous observables (random variables), we show that the properly normalized sums of stationary variables, with respect to the so-called ergodic billiard map, converge in distribution to a totally skewed alpha-stable law, for some alpha between 1 and 2. We also obtain a functional limit theorem to a stable process, as well as to fractional Brownian motion under a multi-particle model. In this talk we will focus on the probabilistic aspects of the proofs.
Homogenization of symmetric nonlocal operators
The talk addresses the homogenization of Lévy-type operators with rapidly oscillating coefficients. We consider cases of periodic and random statistically homogeneous microstructures and show that in the limit we obtain a translation-invariant Lévy-operator. In the periodic case we study both symmetric and non-symmetric kernels whereas in the random case we only investigate symmetric kernels. The talk is based on a recent joint preprint with A. Piatnitski and E. Zhizhina.
Tracy-Widom limit for sparse sample covariance matrices
We consider spectral properties of sample covariance matrices of the form $(\Sigma^{1/2} X)(\Sigma^{1/2} X)^*$, including biadjacency matrices of the bipartite Erdős–Rényi graph. We prove that the rescaled, shifted extremal eigenvalues exhibit GOE Tracy-Widom fluctuations under suitable condition on $\Sigma$ or the sparsity of $X$. We also compute the deterministic shift of the edge from the Marchenko-Pastur law. The talk is based on joint works with Kevin Schnelli and Jong Yun Hwang.
A quantitative theory of the hydrodynamic limit
About a joint work with Deniz Dizdar, Felix Otto, and Tianqi Wu. The hydrodynamic limit is a dynamic manifestation of the law of large numbers. A microscopic random process converges macroscopically to a deterministic process. In this talk, we discuss how to derive quantitative error bounds for the hydrodynamic limit of the Kawasaki dynamics via the two-scale approach. This seems to be the first quantitative statement of this kind.
Generalized couplings
We introduce the concept of a generalized coupling and show how it can be used to show weak uniqueness and ergodicity of stochastic differential equations (with or without delay). This is joint work with Oleg Butkovsky and Alexey Kulik.
Scaling limit of metastable random processes
The metastable random processes exhibit tunneling behavior if there are multiple metastable valleys with the same depth. Tunneling behavior is heuristically described by a Markov chain, but its rigorous formulation is technically cumbersome. We explain recent developments in this sort of problems based on two classic models: the condensing zero-range processes and the small random perturbation of dynamical systems. Even these classic models, the result are very new. This presentation is based on a joint work with Fraydoun Rezakhanlou.
Spaces of algebraic measure trees and triangulations of the circle
Algebraic trees are (continuum) metric trees in which metric distances are ignored. That is, the focus lies on the tree structure only. We show that any (order) separable algebraic tree can be represented by a metric tree. We further consider algebraic measure trees which are additionally equipped with a sampling (probability) measure. This measure gives rise to the branch point distribution which turns out to be the length measure of an intrinsic choice of such a metric tree representation. We will provide a notion of convergence of algebraic measure trees which resembles the idea of the Gromov-weak topology which itself is defined through weak convergence of sample distance matrices. Binary algebraic (measure) trees are of particular interest due to their close connection to triangulations of the circle. We will rely on this connection to show that in the subspace of binary algebraic measure trees Hausdorff convergence of triangulations, weak convergence of sample shapes, sample subtree masses and sample distance matrices are all equivalent. (joint work with Wolfgang Löhr)