Probability

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…

I will sketch why the critical point for percolation on an infinite transitive graph G only depends on the geometry of G on small scales (except in the degenerate case when G is one-dimensional). This is based on joint work with Hutchcroft and was…

Various random graphs models satisfy that each edge appears independently of all other edges but those in a bounded degree graph. Examples include Erdös–Renyi random graphs, random Cayley graphs, random Latin square graphs, and random entangled graphs. We begin the systematic study of random…

I will talk about how critical multiplicative chaos in probability theory is connected to and leads to recent breakthroughs in probabilistic number theory, in particular, the study of random multiplicative functions and character sums. No background in number theory is assumed.

In this paper, we find a natural four-dimensional analog of the moderate deviation results for the capacity of the random walk, which corresponds to Bass, Chen and Rosen's results concerning the volume of the random walk range for dimension 2. We find that the deviation statistics of the…

Diffusion of knowledge models in macroeconomics describe the evolution of an interacting system of agents who perform individual Brownian motions (this is internal innovation) but also can jump on top of each other (this is an agent or a company acquiring knowledge from another agent or company…

In recent years, machine learning has motivated the study of what one might call "nonlinear random matrices." This broad term includes various random matrices whose construction involves the entrywise application of some deterministic nonlinear function, such as ReLU. We study…