Large deviations of spread measures for Gaussian matrices

Abstract

For a large n×m Gaussian matrix, we compute the joint statistics, including large deviation tails, of generalized and total variance-the scaled log-determinant H and trace T of the corresponding n×n covariance matrix. Using a Coulomb gas technique, we find that the Laplace transform of their joint distribution Pn(h, t ) decays for large n, m (with c = m/n ≥ 1 fixed) as Pn(s, w)≈exp(-βn2J(s, w)), where β is the Dyson index of the ensemble and J(s, w) is a β-independent large deviation function, which we compute exactly for any c. The corresponding large deviation functions in real space are worked out and checked with extensive numerical simulations. The results are complemented with a finite n, m treatment based on the Laguerre-Selberg integral. The statistics of atypically small log-determinants is shown to be driven by the split-off of the smallest eigenvalue, leading to an abrupt change in the large deviation speed.