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 positive-semidefinite 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. The methodology provides the solution to an open problem in computational mathematics about sparse random embeddings. The presentation is targeted at a general mathematical audience.

Zoom link: https://upenn.zoom.us/j/98220304722