Gaussian Multiplicative Chaos for i.i.d. matrices

Abstract

We consider $N\times N$ matrices $X$ with independent, identically distributed entries, and prove that the sequence of measures $\frac{ | \det (X-z)|^γ}{\mathbb{E}[ | \det (X-z)|^γ]}$ converge to the Gaussian Multiplicative Chaos in the full subcritical regime $γ\in (0, 2 \sqrt{2})$ as $N \to \infty$. Our result holds for both symmetry classes and in particular is new even for real Ginibre matrices, and is the first such convergence for any non-invariant ensemble of random matrices. We also 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.