Current Series

[View Past Series]

Fri Sep 13

Probability Seminar

2:30pm - Vincent Hall 311
Critical behavior for percolation on graphs with given degrees
Souvik Dhara, MIT

We discuss critical behavior of percolation on finite random networks. In a seminal paper, Aldous (1997) identified the scaling limit for the component sizes in the critical window of phase transition for the Erdos-Renyi random graph (ERRG). Subsequently, there has been a surge in the literature, revealing several interesting scaling limits of these critical components, namely, the component size, diameter, or the component itself when viewed as a metric space. Fascinatingly, when the third moment of the asymptotic degree distribution is finite, many random graph models have been shown to exhibit a universality phenomenon in the sense that their scaling exponents and limit laws are the same as the ERRG. In contrast, when the asymptotic degree distribution is heavy-tailed (having an infinite third moment), the limit law turns out to be fundamentally different from the ERRG case and in particular, becomes sensitive to the precise asymptotics of the highest degree vertices.

In this talk, we will focus on random graphs with a prescribed degree sequence. We start by discussing recent scaling limit results, and explore the universality classes that arise from heavy-tailed networks. Of particular interest is a new universality class that arises when the asymptotic degree distribution has an infinite second moment. Not only it gives rise to a completely new universality class, it also exhibits several surprising features that have never been observed in any other universality class so far.

This is based on joint works with Shankar Bhamidi, Remco van der Hofstad, Johan van Leeuwaarden and Sanchayan Sen.

Fri Sep 27

Probability Seminar

2:30pm - Vincent Hall 311
Random matrices, operators and analytic functions
Benedek Valko, UW Madison

The finite circular beta-ensembles and their point process scaling limit can be represented as the spectra of certain random differential operators. These operators can be realized on a single probability space so that the point process scaling limit is a consequence of an operator level limit. The construction allows the derivation of the scaling limit of the normalized characteristic polynomials of the finite models to a random analytic function. I will review these representations and constructions, and present a couple of applications.

Joint with B. Virág (Toronto).

Fri Oct 04

Probability Seminar

2:30pm - Vincent Hall 311
Reconstruction problems on amenable graphs
Ahmed El Alaoui, Stanford University

We consider the problem of reconstructing a hidden assignment of random variables (x_1,…, x_n) sitting on the nodes of a graph, given noisy observations of pairs (x_u, x_v) for every edge (u,v) of the graph. Such problems have been extensively studied when the underlying graph is ‘mean-field’ (e.g., the complete graph, Erdos-Renyi, random regular,…), in which case the existence of a ``possible-but-hard’’ phase where reconstruction is possible but computationally difficult is ubiquitous.

In contrast, the picture is dramatically different when the graph is amenable. I will represent a generic result about the optimality of local algorithms for computing marginals under a somewhat strong model of side information. Next, I will focus on Z_2 synchronization (aka planted Ising model) on the Euclidean lattice with weaker side information, and discuss a renormalization-based procedure for reconstructing the assignment.

Joint work with Andrea Montanari.

Fri Oct 04

Probability Seminar

3:30pm - Vincent Hall 311
Invertibility of inhomogenuous heavy-tailed matrices
Galyna Livshyts, Georgia Tech

We will show the sharp estimate on the behavior of the smallest singular value of random matrices under very general assumptions. One of the key steps in the proof is a result about the efficient discretization of the unit sphere in an n-dimensional euclidean space. This result allows us to work with very general random matrices. The proof of the result will be outlined. Partially based on the joint work with Tikhomirov and Vershynin.

Fri Oct 25

Probability Seminar

2:30pm - Vincent Hall 311
Probability Seminar
Kevin Leder, UMN
Fri Nov 08

Probability Seminar

2:30pm - Vincent Hall 311
Probability Seminar
Yunpeng Shi, UMN
Fri Nov 15

Probability Seminar

2:30pm - Vincent Hall 311
Probability Seminar
Anna Kraut, Bonn