Large deviations of spread measures for Gaussian matrices

Abstract

For a large $n\times m$ 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 $n\times n$ covariance matrix. Using a Coulomb gas technique, we find that the Laplace transform of their joint distribution ${\mathcal{P}}_n(h,t)$ decays for large $n,m$ (with $c=m/n\ge 1$ fixed) as ${\hat{\mathcal{P}}}_n(s,w)\approx \exp \left(-\beta n^{2}J(s,w)\right)$, where $\beta$ is the Dyson index of the ensemble and $J(s,w)$ is a $\beta$-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.

20 pages, 3 figures. v4: final version

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