Probability

Branching Brownian motion (BBM) is a classical probabilistic model that has "log-correlated" behavior. Its limiting extremal process has been derived to be that of a randomly shifted clustered Poisson point process with an exponential intensity (Aidekon-Berestycki-Brunet-Shi; Arguin-Bovier-…

We show that the shortest s-t path problem has the overlap-gap property in (i) sparse G(n,p) graphs and (ii) complete graphs with i.i.d. exponential edge weights. Furthermore, we demonstrate that in sparse G(n,p) graphs, shortest path is solved by O(log n)-degree polynomial estimators, and a…

Given a Gaussian energy function H(x) and another random process O(x) (an observable) both defined on the same configuration space, what is the law of O(y) for y sampled from the Gibbs measure associated to H(x)? We will see the answer to this question in the high-temperature phase in a general…

A self-interacting random walk is a random process evolving in an environment which depends on its history. In this talk, we will discuss a few examples of these walks including the Lorentz gas, the mirror walk and the cyclic walk in the interchange process. I will present a method to analyze…

Recently, there has been much progress in understanding stationary measures for colored (also called multi-species or multi-type) interacting particle systems. In this work, we present a unified approach to constructing stationary measures for several colored particle systems on the ring and the…

While the notion of spectral gap is a fundamental and very useful feature of reversible Markov chains, there is no standard analogue of this notion for non-reversible chains. In this talk I will present a simple proposal for spectral gap of non-reversible chains and show that it shares all the…

Random regular graphs form a ubiquitous model for chaotic systems. However, the spectral properties of their adjacency matrices have proven difficult to analyze because of the strong dependence between different entries. In this talk, I will describe recent work that shows that despite this, the…

The two-dimensional one-component plasma (OCP), also known as the Coulomb gas, is a system that consists of identical electrically charged particles embedded in a uniform background of the opposite charge, interacting through a logarithmic potential and kept at a fixed temperature. In the 1990s…

The extremal process of branching Brownian motion (BBM) — i.e., the collection of particles farthest from the origin — has gained lots of attention in dimension d = 1 due to its significance to the universality class of log-correlated fields, as well as to certain PDEs. In recent years, a…

Trees are everywhere in applied probability and computer science. It is natural to ask about what a typical tree looks like. I will review a surprisingly large literature. For example, Cayley's theorem tells us there are $n^{(n-2)}$ labeled trees and it's easy to work with them. There is no…