Seminars and Colloquia by Series

Dynamical critical 2d first-passage percolation

Series
Stochastics Seminar
Time
Thursday, March 3, 2022 - 15:30 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
David HarperGeorgia Tech

In first-passage percolation (FPP), we let \tau_v be i.i.d. nonnegative weights on the vertices of a graph and study the weight of the minimal path between distant vertices. If F is the distribution function of \tau_v, there are different regimes: if F(0) is small, this weight typically grows like a linear function of the distance, and when F(0) is large, the weight is typically of order one. In between these is the critical regime in which the weight can diverge, but does so sublinearly. This talk will consider a dynamical version of critical FPP on the triangular lattice where vertices resample their weights according to independent rate-one Poisson processes. We will discuss results which show that if sum of F^{-1}(1/2+1/2^k) diverges, then a.s. there are exceptional times at which the weight grows atypically, but if sum of k^{7/8} F^{-1}(1/2+1/2^k) converges, then a.s. there are no such times. Furthermore, in the former case, we compute the Hausdorff and Minkowski dimensions of the exceptional set and show that they can be but need not be equal. These results show a wider range of dynamical behavior than one sees in subcritical (usual) FPP. This is a joint work with M. Damron, J. Hanson, W.-K. Lam.

This talk will be given on Bluejeans at the link https://bluejeans.com/283104959/2281

Phase transitions in soft random geometric graphs

Series
Stochastics Seminar
Time
Thursday, January 13, 2022 - 15:30 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/257822708/6700
Speaker
Suqi LiuPrinceton University

Random graphs with latent geometric structure, where the edges are generated depending on some hidden random vectors, find broad applications in the real world, including social networks, wireless communications, and biological networks. As a first step to understand these models, the question of when they are different from random graphs with independent edges, i.e., Erd\H{o}s--R\'enyi graphs, has been studied recently. It was shown that geometry in these graphs is lost when the dimension of the latent space becomes large. In this talk, we focus on the case when there exist different notions of noise in the geometric graphs, and we show that there is a trade-off between dimensionality and noise in detecting geometry in the random graphs.

Statistical and computational limits for sparse graph alignment

Series
Stochastics Seminar
Time
Thursday, December 9, 2021 - 15:30 for 1 hour (actually 50 minutes)
Location
Online
Speaker
Luca GanassaliINRIA

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

Efficient Volatility Estimation Of Lévy Processes of Unbounded Variation

Series
Stochastics Seminar
Time
Thursday, November 11, 2021 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
José Figueroa-LópezWashington University in St. Louis

Statistical inference of stochastic processes based on high-frequency observations has been an active research area for more than a decade. The most studied problem is the estimation of the quadratic variation of an Itô semimartingale with jumps. Several rate- and variance-efficient estimators have been proposed when the jump component is of bounded variation. However, to date, very few methods can deal with jumps of unbounded variation. By developing new high-order expansions of truncated moments of Lévy processes, a new efficient estimator is developed for a class of Lévy processes of unbounded variation. The proposed method is based on an iterative debiasing procedure of truncated realized quadratic variations. This is joint work with Cooper Bonience and Yuchen Han.

Gibbsian line ensembles and beta-corners processes

Series
Stochastics Seminar
Time
Thursday, November 4, 2021 - 16:30 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Evgeni DimitrovColumbia University

Please Note: The link for the talk is https://bluejeans.com/492736052/2047

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.

Many nodal domains in random regular graphs

Series
Stochastics Seminar
Time
Thursday, October 28, 2021 - 15:30 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Theo McKenzieBerkeley

If we partition a graph according to the positive and negative components of an eigenvector of the adjacency matrix, the resulting connected subcomponents are called nodal domains. Examining the structure of nodal domains has been used for more than 150 years to deduce properties of eigenfunctions. Dekel, Lee, and Linial observed that according to simulations, most eigenvectors of the adjacency matrix of random regular graphs have many nodal domains, unlike dense Erdős-Rényi graphs. In this talk, we show that for the most negative eigenvalues of the adjacency matrix of a random regular graph, there is an almost linear number of nodal domains. Joint work with Shirshendu Ganguly, Sidhanth Mohanty, and Nikhil Srivastava.

Towards robust and efficient mean estimation

Series
Stochastics Seminar
Time
Thursday, September 16, 2021 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Stas MinskerUniversity of Southern California

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.

Learning Gaussian mixtures with algebraic structure

Series
Stochastics Seminar
Time
Thursday, April 22, 2021 - 15:30 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/129119189
Speaker
Victor-Emmanuel BrunelENSAE/CREST

We will consider a model of mixtures of Gaussian distributions, called Multi-Reference Alignment, which has been motivated by imaging techniques in chemistry. In that model, the centers are all related with each other by the action of a (known) group of isometries. In other words, each observation is a noisy version of an isometric transformation of some fixed vector, where the isometric transformation is taken at random from some group of isometries and is not observed. Our goal is to learn that fixed vector, whose orbit by the action of the group determines the set of centers of the mixture. First, we will discuss the asymptotic performances of the maximum-likelihood estimator, exhibiting two scenarios that yield different rates. We will then move on to a non-asymptotic, minimax approach of the problem.

The parking model in Z^d

Series
Stochastics Seminar
Time
Thursday, April 15, 2021 - 15:30 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
David SivakoffThe Ohio State University

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 

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