Georgia Institute of Technology College of Sciences

Graph alignment refers to recovering the underlying vertex correspondence between two random graphs with correlated edges. This problem can be viewed as an average-case and noisy version of the well-known graph isomorphism problem. For correlated Erdős-Rényi random graphs, we will give insights on the fundamental limits for the planted formulation of this problem, establishing statistical thresholds for partial recovery. From the computational point of view, we are interested in designing and analyzing efficient (polynomial-time) algorithms to recover efficiently the underlying alignment: in a sparse regime, we exhibit an local rephrasing of the planted alignment problem as the correlation detection problem in trees. Analyzing this related problem enables to derive a message-passing algorithm for our initial task and gives insights on the existence of a hard phase.

Based on joint works with Laurent Massoulié and Marc Lelarge: 

https://arxiv.org/abs/2002.01258

https://arxiv.org/abs/2102.02685

https://arxiv.org/abs/2107.07623

Gibbs measures are ubiquitous in statistical mechanics and probability theory. In this talk I will discuss two types of classes of Gibbs measures – random line ensembles and triangular particle arrays, which have received considerable attention due, in part, to their occurrence in integrable probability.
Gibbsian line ensembles can be thought of as collections of finite or countably infinite independent random walkers whose distribution is reweighed by the sum of local interactions between the walkers. I will discuss some recent progress in the asymptotic study of Gibbsian line ensembles, summarizing some joint works with Barraquand, Corwin, Matetski, Wu and others.
Beta-corners processes are Gibbs measures on triangular arrays of interacting particles and can be thought of as analogues/extensions of multi-level spectral measures of random matrices. I will discuss some recent progress on establishing the global asymptotic behavior of beta-corners processes, summarizing some joint works with Das and Knizel.

Several constructions of the estimators of the mean of a random variable that admit sub-Gaussian deviation guarantees and are robust to adversarial contamination under minimal assumptions have been suggested in the literature. The goal of this talk is to discuss the size of constants appearing in the bounds, both asymptotic and non-asymptotic, satisfied by the median-of-means estimator and its analogues. We will describe a permutation-invariant version of the median-of-means estimator and show that it is asymptotically efficient, unlike its “standard" version. Finally, applications and extensions of these results to robust empirical risk minimization will be discussed.

At each site of Z^d, initially there is a car with probability p or a vacant parking spot with probability (1-p), and the choice is independent for all sites. Cars perform independent simple, symmetric random walks, which do not interact directly with one another, and parking spots do not move. When a car enters a site that contains a vacant spot, then the car parks at the spot and the spot is filled – both the car and the spot are removed from the system, and other cars can move freely through the site. This model exhibits a phase transition at p=1/2: all cars park almost surely if and only if p\le 1/2, and all vacant spots are filled almost surely if and only if p \ge 1/2. We study the rates of decay of cars and vacant spots at, below and above p=1/2. In many cases these rates agree with earlier findings of Bramson—Lebowitz for two-type annihilating systems wherein both particle types perform random walks at equal speeds, though we identify significantly different behavior when p<1/2. Based on joint works with Damron, Gravner, Johnson, Junge and Lyu.

Online at https://bluejeans.com/129119189