Probability
Organizers: Amir Dembo (Autumn) & Eric Thoma (Spring)
Past Events
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…
We study the branching random walk under a "hard wall constraint", namely when the heights of all particles in the most recent generation are conditioned to be positive. We obtain sharp asymptotics for the probability of this event and for various statistics, conditional on its occurrence. In…
The binary perceptron problem asks us to find a sign vector in the intersection of independently chosen random halfspaces with a fixed intercept. The computational landscape of the binary perceptron is not yet well-understood. In some regimes there may be an information-computation gap, but…
While unimodal probability distributions are well understood in dimension 1, the same cannot be said in high dimension without imposing stronger conditions such as log-concavity. I will explain a new approach to proving confinement (e.g., variance upper bounds) for high-dimensional unimodal…
For rotationally invariant first passage percolation on the plane, we use a multi-scale argument to prove stretched exponential concentration of the passage times at the scale of the standard deviation. Our results are proved for several standard rotationally invariant models of first passage…