Asymptotic behavior of random determinants in the Laguerre, Gram and Jacobi ensembles

We consider properties of determinants of some random symmetric matrices issued from multivariate statistics: Wishart/Laguerre ensemble (sample covariance matrices), Uniform Gram ensemble (sample correlation matrices) and Jacobi ensemble (MANOVA). If n is the size of the sample, r ≤ n the number of variates and Xn,r such a matrix, a generalization of the Bartlett-type theorems gives a decomposition of detXn,r into a product of r independent gamma or beta random variables. For n fixed, we study the evolution as r grows, and then take the limit of large r and n with r/n = t ≤ 1. We derive limit theorems for the sequence of processes with independent increments {n−1 log detXn,⌊nt⌋, t ∈ [0, T ]}n for T ≤ 1 : convergence in probability, invariance principle, large deviations. Since the logarithm of the determinant is a linear statistic of the empirical spectral distribution, we connect the results for marginals (fixed t) with those obtained by the spectral method. Actually, all the results hold true for log gases or β models, if we define the determinant as the product of charges. The classical matrix models (real, complex, and quaternionic) correspond to the particular values β = 1, 2, 4 of the Dyson parameter.