Probability
Organizers: Amir Dembo (Autumn) & Eric Thoma (Spring)
Upcoming Events
I shall discuss first-passage percolation on Cayley graphs of Gromov hyperbolic groups under mild conditions on the passage time distribution. Appealing to deep geometric and topological facts about hyperbolic groups and their boundaries, several questions become more tractable in this set-up…
Past Events
The (d+1)D solid-on-solid model is a simple model of integer-valued height functions that approximates the low-temperature interface of an Ising model. When $d\geq 2$, with zero-boundary conditions, at low temperatures the surface is localized about height 0, but when constrained to take only…
Let A be an n by n matrix with iid Ber(d/n) entries. We show that the empirical measure of the eigenvalues converges, in probability, to a deterministic distribution. The proof involves incrementally exposing the randomness of the underlying matrix and studying the evolution of the singular…
Lévy matrices are symmetric random matrices whose entries are in the domain of attraction of an \alpha stable law. For \alpha < 1, it had been predicted that these matrices exhibit an Anderson transition, also called a mobility edge, a point in the spectrum where eigenvector behavior sharply…
Optimizing high-dimensional functions generated from random data is a central problem in modern statistics and machine learning. As these objectives are highly non-convex, the maximum value reachable by efficient algorithms is usually smaller than the maximum value that exists, and…
The Coulomb gas is a statistical physics model consisting of N particles interacting with electrostatic repulsion and with a global confining potential. I will show how a certain subharmonic structure associated with the k-point function arises. This structure implies new bounds on quantities…
Last May I spoke about a series of multi-scale arguments to establish scaling relations for rotationally invariant first passage percolation in the plane. In this talk I will discuss how these methods can be used to establish the chaotic nature of the optimal path, that is, after a small…
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…