Characteristic polynomial of self-normalized random matrices

Abstract

We study the characteristic polynomial of self-normalized random matrices, whose rows are independent and normalized to have unit $\mathrm{L}^2$ norm. The entries before normalization are assumed to have regularly varying tails with tail index $α\in [0,2]$. We prove that, outside the unit disk, the characteristic polynomial converges to a random analytic function $F_α$. We identify $F_α$ as a multiplicative chaos described in terms of Poisson point processes. The family of limiting functions $(F_α)_{α\in[0,2]}$ interpolates between two universal regimes: Poisson multiplicative chaos at $α=0$ and Gaussian multiplicative chaos at the boundary $α=2$. Thus, self-normalization provides a matrix model where one observes the transition between Poissonian and Gaussian regimes for the limiting characteristic polynomial as the tail index varies. A similar transition is found for the fluctuations of the traces of self-normalized matrices. As an application of our results, we derive that the spectral radius of self-normalized matrices is asymptotically bounded above by one in probability for any symmetric entry distribution.