Abstract: The connection between random matrices and Gaussian multiplicative chaos (GMC) has been studied intensively since the work of Fyodorov and Keating (2014). There has been recent interest in the appearance of GMC in dynamical ensembles of random matrices and their eigenvalues, such as the Dyson Brownian motion (Keles 2025). We propose a Riemann-Hilbert approach to this problem. I will explain how our approach allows one to probe the “microscopic scale” (a scale which is inaccessible via loop equations), where the appearance of Painlevé transcendents is anticipated. This is joint work with Ahmad Barhoumi.