Probability

In this talk, I will discuss recent progress on disordered quantum systems beyond the mean-field regime. Earlier this year (joint with H.-T. Yau), we resolved the delocalization conjecture for random band matrices by developing the loop hierarchy method and its tree approximation. In our new…

In this talk, we will consider the model of random walks in a space-time random environment, which can be thought of as a discrete model for diffusing particles in a time-dependent random medium. We will study the scaling limits of these models in certain moderate deviation scaling regimes for…

The cavity method is a powerful, non-rigorous technique at the heart of the theory of spin glasses. Its application to dynamics, called the dynamical cavity method, has been one of the main tools in the physicist's toolkit for describing the asymptotics of algorithms in disordered systems. One…

We will introduce the Young generating function and use it to characterize the law of large numbers and the central limit theorem behaviors for random partitions. As an application of these results, we present a framework to obtain conditional Gaussian Free Field fluctuations for height…

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…