# Seminars & Events for Probability Seminar

##### Local Universality of Random Functions

In this talk, we discuss local universality results for a general class of random functions that includes random trigonometric polynomials and random orthogonal polynomials. We then apply these results to obtain estimates for the number of real roots and prove, in some cases, that this number satisfies the Central Limit Theorem. This is joint work with Van Vu.

##### TBA-Lionel Levine

##### Random interlacements and the Gaboriau-Lyons problem

Given a Cayley graph for a non-amenable group, can one find a factor of an IID process that gives a random forest in which the average degree is greater than 2? Gaboriau-Lyons proved that the answer is `yes’ and this has interesting applications to ergodic theory. However, they required a lower bound on the entropy of the IID process. I’ll show, via random interlacements, how to remove this restriction. One application is that every IID process is a factor of every (nontrivial) IID process.

##### Cover time of trees and of the two dimensional sphere

I will begin by reviewing the general relations that exist between the cover time of graphs by random walk and the Gaussian free field on the graph, and explain the strength and limitations of these general methods. I will then discuss recent results concerning the cover time of the binary tree of depth $n$ by simple random walk, and in particular sharp fluctuation results for the cover time, mirroring those for the maximal displacement of branching random walk; certain barrier estimates for Bessel processes play a crucial role. Finally, I will describe how these technique can be applied to the study of the cover time of the 2-sphere by the Brownian sausage.

##### The KPZ fixed point

The KPZ universality class contains one dimensional growth models,