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.

98 pages, added GMC on the real line for real matrices, references added