Simon Coste -- Convergence of the characteristic polynomial of sparse matrices
Channel: Matrices Et Graphes Aléatoires GDR CNRS
Date: 2021-12-09

Abstract: The reverse characteristic polynomial of a matrix A is q(z)=det(I - z A). I will show that when A is a (non-Hermitian) n x n random matrix with iid Bernoulli(d/n) entries, with d fixed, then when n goes to infinity, the polynomial q converges towards a random infinite holomorphic function F, which is the exponential of a random series with Poisson coefficients. This entails a short proof of previous results on the asymptotics of the eigenvalues of A, a strategy inspired by a recent paper by Bordenave, Chafai and Garcia-Zelada. The function F is itself a Poisson analog of the Gaussian Holomorphic Chaos, and I will list some questions regarding its coefficients and its boundary behaviour.