Gaussian Multiplicative Chaos for i.i.d. matrices

We consider $N\times N$ matrices $X$ with independent, identically distributed entries, and prove that the sequence of measures $\frac{\mathrm{\mid }\det (X-z){\mathrm{\mid }}^{\gamma }}{\mathbb{E}[\mathrm{\mid }\det (X-z){\mathrm{\mid }}^{\gamma }]}$ converge to the Gaussian Multiplicative Chaos in the full subcritical regime $\gamma \in (0,2\sqrt{2})$ as $N\to \mathrm{\infty }$. 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 $\mathrm{\mid }\det (X-z)\mathrm{\mid }$ 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.