Mathematics

The Dyson Brownian motion (DMB) is a system of infinitely many interacting Brownian motions with logarithmic interaction potential, which was introduced by Freeman Dyson '62 in relation to the random matrix theory. In this talk, we show that an infinite-dimensional differential structure induced by the DBM has a Bakry-Émery lower Ricci curvature bound. As an application, we show that the DBM (with arbitrary inverse temperature beta >0) can be realised as the unique Wasserstein-type gradient flow of the Boltzmann-Shannon entropy associated with sine_beta ensemble. I will start with a gentle introduction of the field. If time allows, I will also explain several open questions related to (extended) metric measure geometry, random matrices and stochastic analysis.