Positive random walks and random psd matrices
Positive random walks and random psd matrices

Joel Tropp, Caltech
Fine Hall 214
On the real line, a random walk that can only move in the positive direction is very unlikely to remain close to its starting point. After a fixed number of steps, the left tail has a Gaussian profile, under minimal assumptions. Remarkably, the same phenomenon occurs when we consider a positive random walk on the cone of positivesemidefinite matrices. After a fixed number of steps, the minimum eigenvalue is also described by a Gaussian model.
This talk introduces a new way to make this intuition rigorous, and it presents an application to the sample covariance matrix of a very sparse vector. The key technical ingredient is Stahl's theorem (2012), recently improved by Otte Heinävaara (PhD Princeton, 2024).