Large deviations of spread measures for Gaussian matrices

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 $c=m\mathrm{/}n\ge 1$ fixed) as $\hat{$\mathcal{P}$}_n(s,w)\approx \exp\left(-\beta n^2 J(s,w)\right)$,where$\beta$istheDysonindexoftheensembleand$J(s,w)$isa$\beta$-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.