Probability

The study of random holonomies (or Wilson loops) of the 2D Yang-Mills model goes back to the late 1980s. The law of these loop observables can be described in terms of heat kernels on Lie groups. In this talk, we start with an introductory review of these ideas. Then we discuss our new result in…

Critical holomorphic multiplicative chaos (HMC) arises naturally from the studies of characteristic functions of CUE and partial sums of random multiplicative functions. We investigate the low moments of secular (Fourier) coefficients of the critical HMC. We establish:

1. universality…

Exponential last passage percolation (LPP) is a canonical planar directed model of random geometry in the KPZ universality class where the Euclidean metric is distorted by i.i.d. noise. One can also consider a dynamical version of LPP, where the noise is resampled at a constant rate, thereby…

Nearest-neighbor Bernoulli percolation in Z^d has an upper critical dimension, above which several features of the model, including critical exponents, become dimension independent. Unlike in intermediate dimensions between 2 and 6, there has been a lot of progress on the high-dimensional case.…

We consider favorite sites, i.e., sites that achieve the maximal local time for a discrete time simple random walk. We show that the limsup of the number of favorite sites is 3 with probability one in d=2. We also give sharp asymptotics of the number in higher dimensions.

This talk is…

Random band matrices have entries concentrated in a narrow band of width W around the main diagonal, modeling systems with spatially localized interactions. We consider one-dimensional random band matrices with bandwidth W >> N^½, general variance profile, and arbitrary entry distributions…

The probability community has obtained fruitful results about the connectivity of random graphs in the last 50 years. Random topology is an emerging field that studies higher-order connectivity of random simplicial complexes, which are higher-order generalizations of graphs. Many classical…

Given a discrete-time non-lattice supercritical branching random walk (or branching Brownian motion) in $\mathbb{R}^d$, we investigate its first passage time to a shifted unit ball of a distance $x$ from the origin, conditioned upon survival. We provide precise asymptotics up to $O(1)$ (…

Motivated by modeling time-dependent processes on networks like social interactions and infection spread, we consider a version of the classical Erdős–Rényi random graph G(n,p) where each edge has a distinct random timestamp, and connectivity is constrained to sequences of edges with increasing…

Spin models on graphs are a source of many interesting questions in statistical physics, algorithms, and combinatorics. The Ising model is a classical example: first introduced as a model of magnetization, it can combinatorially be described as a weighted probability distribution on two-vertex…