Gaussian Multiplicative Chaos for i.i.d. matrices

Abstract

We consider $N\times N$ matrices $X$ with i.i.d. entries, and prove that the random measure $\mu_N(z):=\frac{ | \det (X-z)|^\gamma}{\mathbb{E}[ | \det (X-z)|^\gamma]} \mathrm{d} z\mathrm{d} \overline{z}$ converges to the Gaussian Multiplicative Chaos (GMC) on the unit disc in the full subcritical regime $\gamma \in (0, 2 \sqrt{2})$ as $N \to \infty$. Our result holds for both symmetry classes and in particular is new even for real Ginibre matrices. This result is the first of its kind for any non-invariant ensemble of random matrices. In the case of real matrices, we further prove that the restriction of $\mu_N(z)$ to the unit interval converges to a one-dimensional GMC. We establish the asymptotics for the $K$-point function of $| \det (X-z)|$ at any collection of mesoscopically separated points $z_i$. Our methods are analytic and probabilistic in nature, relying in part on the dynamical approach based on Dyson Brownian motion.