I am co-organizing a virtual Probability Seminar with Wei-Kuo Chen and Arnab Sen at University of Minnesota for
the Spring 2024 semester. Please email me to get the Zoom link for the seminar series.
The seminar is held on Fridays at 2:30pm (ET), unless otherwise noted below.
| 02/02/2024 | Devraj Duggal (UMN) |
| Title: On Spherical Covariance Representations. | |
| Abstract: We first motivate the study of covariance representations by surveying preceding results in the Gauss space. Their spherical counterparts are then derived thereby allowing applications to the spherical concentration phenomenon. The applications include second order concentration inequalities. This talk is based on joint work with Sergey Bobkov. | |
| 02/09/2024 | Xiao Shen (Utah) |
| Title: Random growth models and the KPZ universality. | |
| Abstract: Many two-dimensional random growth models, including first- and last-passage percolation, are conjectured to fall within the KPZ universality class under mild assumptions on the underlying noise. In recent years, researchers have focused on a subset of exactly solvable models, where these conjectures can be rigorously verified. A wide array of methods has been employed, encompassing integrable probability, Gibbsian line ensemble, percolation arguments, and coupling techniques. This talk discusses a specific line of research that combines percolation arguments and coupling techniques to gain insights into the random geometry and space-time profiles of such growth models within the KPZ class. | |
| 02/16/2024 | Ratul Biswas (UMN) |
| Title: Limiting eigenvalue distribution of heavy-tailed Toeplitz matrices. | |
| Abstract: We show that under an appropriate scaling, the limiting eigenvalue distribution of a symmetric Toeplitz matrix with i.i.d. entries drawn from an α-stable distribution (0 < α < 2) converges weakly to a random symmetric probability distribution on R. We express this random probability distribution in terms of the spectral measure of a random unbounded operator on 𝓁 2(Z) and study some properties of this distribution for different values of α. Based on joint work with Arnab Sen. | |
| 02/23/2024 | Bhaswar B. Bhattacharya (UPenn) |
| Title: Higher-Order Graphon Theory: Fluctuations, Inference, and Degeneracies. | |
| Abstract: Motifs (patterns of subgraphs), such as edges and triangles, encode important structural information about the geometry of a network. Consequently, counting motifs in a large network is an important statistical and computational problem. In this talk we will consider the problem of estimating motif densities and fluctuations of subgraph counts in an inhomogeneous random graph sampled from a graphon. We will show that the limiting distributions of subgraph counts can be Gaussian or non-Gaussian, depending on a notion of regularity of subgraphs with respect to the graphon. Using these results and a novel multiplier bootstrap for graphons, we will construct joint confidence sets for the motif densities. Finally, we will discuss various structure theorems and open questions about degeneracies of the limiting distribution. (Joint work with Anirban Chatterjee, Soham Dan, and Svante Janson.) | |
| 03/01/2024 | Youngtak Sohn (MIT) |
| Title: Phase transitions of random constraint satisfaction problems. | |
| Abstract: The framework of constraint satisfaction problems (CSPs) captures many fundamental problems in combinatorics and computer science, such as finding a proper coloring of a graph or solving Boolean satisfiability problems. To study the typical cases of CSPs, statistical physicists have proposed a detailed picture of the solution space for random CSPs based on non-rigorous methods from spin glass theory. In this talk, I will first survey the conjectured rich phase diagrams of random CSPs in the one-step replica symmetry breaking universality class. Then, I will describe the recent progress in understanding the global and local geometry of solutions, particularly in random regular NAE-SAT problem. This is based on joint works with Danny Nam and Allan Sly. | |
| 03/08/2024 | Spring Break at UMN |
| 03/15/2024 | Spring Break at LU and SSP2024 (Mar. 13-16 at Rice) |
| 03/22/2024 | Lingfu Zhang (Berkeley) |
| Title: Random Lozenge Tiling and Universality of the Pearcey Process. | |
| Abstract: It has been known since Cohn-Kenyon-Propp (2000) that uniformly random tiling by lozenges exhibits frozen and disordered regions, which are separated by the 'arctic curve'. For a generic simply connected polygonal domain, the microscopic statistics are widely predicted to be universal, being one of (1) discrete sine process inside the disordered region (2) Airy line ensemble around a smooth point of the curve (3) Pearcey process around a cusp of the curve (4) GUE corner process around a tangent point of the curve. These statistics were proved years ago for special domains, using exact formulas; as for universality, much progress was made more recently. In this talk, I will present proof of the universality of (3), the remaining open case. Our approach is via a refined comparison between tiling and non-intersecting random walks, for which a new universality result of the Pearcey process is also proved. This is joint work with Jiaoyang Huang and Fan Yang. | |
| 03/29/2024 | Subhajit Goswami (TIFR, India) Note: special time! |
| at 11am | Title: Percolation phase transition for the vacant set of random walks and random interlacements. |
| Abstract: The random interlacements on the hypercubic lattice in dimensions 3 and higher (or any transient graph for that matter) describe the trace of a Poisson ensemble of bi-infinite random walk trajectories indexed by a time-like intensity parameter. The model was introduced in a seminal paper by Sznitman as a local limit of the trace of simple random walk on the (discrete) torus when it runs for an amount of time proportional to volume of the torus. This immediately gives rise to an intriguing percolation problem where one is interested in the percolative properties of the vacant set or equivalently the question of disconnection from the boundary by random walk trajectories. Often referred to as the fragmentation of a torus by random walk, this is a prototypical example of a percolation model showing non-local dependence and one which is much more "rigid" in several aspects compared to the Bernoulli percolation, the random-cluster models and even the closely related level-sets of Gaussian free field. In this talk we will review some very recent developments on the anatomy of the clusters of the vacant set of random interlacements and random walks. These involve ideas ranging from the potential theory of random walks to some very recent advances in Percolation theory. Based on several (including some upcoming) joint works with Hugo Dumini-Copin, Pierre-Francois Rodriguez, Franco Severo, Augusto Teixeira and Yuriy Shulzhenko. | |
| 04/05/2024 | Sayan Das (UChicago) |
| Title: Weak Universality in Random Walks in Random Environments. | |
| Abstract: We consider one dimensional simple random walks whose all one step transition probabilities are iid [0,1]-valued mean 1/2 random variables. In this talk, we will explain how under a certain moderate deviation scaling the quenched density of the walk converges weakly to Stochastic Heat Equation with multiplicative noise. Our result captures universality in the sense that it holds for all non-trivial laws for random environments. Time permitting, we will discuss briefly how our proof techniques depart from the existing techniques in literature. Based on a joint work with Hindy Drillick and Shalin Parekh. | |
| 04/12/2024 | Zhichao Wang (UCSD) |
| Title: Nonlinear spiked covariance matrices and signal propagation in neural network models. | |
| Abstract: In this talk, I will first present some recent work for the extreme eigenvalues of sample covariance matrices with spiked population covariance. Extending previous random matrix theory, we will characterize the spiked eigenvalues outside the bulk distribution and their corresponding eigenvectors for a nonlinear version of the spiked covariance model. Then, we will apply this new result to deep neural network models. Many recent works have studied the eigenvalue spectrum of the Conjugate Kernel (CK) defined by the nonlinear feature map of a feedforward neural network. However, existing results only establish weak convergence of the empirical eigenvalue distribution and fall short of providing precise quantitative characterizations of the ''spike'' eigenvalues and eigenvectors that often capture the low-dimensional signal structure of the learning problem. Using our general result for spiked sample covariance matrices, we will give a quantitative description of how spiked eigenstructure in the input data propagates through the hidden layers of a neural network with random weights. As a second application, we can study a simple regime of representation learning where the weight matrix develops a rank-one signal component over gradient descent training and characterize the alignment of the target function with the spike eigenvector of the CK on test data. This analysis will show how neural networks learn useful features at the early stage of training. This is a joint work with Denny Wu and Zhou Fan. | |
| 04/26/2024 | Dor Elboim (IAS) |
| Title: Poisson-Dirichlet distribution for the interchange process in five dimensions. | |
| Abstract: In the interchange process on a graph G=(V,E), distinguished particles are placed on the vertices of G with independent Poisson clocks on the edges. When the clock of an edge rings, the two particles on the two sides of the edge interchange. In this way, a random permutation πβ: V → V is formed for any time β > 0. One of the main objects of study is the cycle structure of the random permutation and the emergence of long cycles. We consider the process on the torus of side length L in dimension d ≥ 5 and prove that macroscopic cycles emerge after a long time β. These are cycles whose length is proportional to the volume of the torus Ld. Moreover, we show that the cycle lengths converge to the Poisson-Dirichlet distribution. This is a joint work with Allan Sly. | |