Pathwise asymptotic behavior of random determinants in the uniform Gram and Wishart ensembles

Abstract

This paper concentrates on asymptotic properties of determinants of some random symmetric matrices. If B_{n,r} is a n x r rectangular matrix and B_{n,r}' its transpose, we study det (B_{n,r}'B_{n,r}) when n,r tends to infinity with r/n \to c\in (0,1). The r column vectors of B_{n,r} are chosen independently, with common distribution \nu_n. The Wishart ensemble corresponds to \nu_n = {\cal N}(0, I_n), the standard normal distribution. We call uniform Gram ensemble the ensemble corresponding to \nu_n = \sigma_n, the uniform distribution on the unit sphere `S_{n-1}. In the Wishart ensemble, a well known Bartlett's theorem decomposes the above determinant into a product of chi-square variables. The same holds in the uniform Gram ensemble. This allows us to study the process \{\frac{1}{n}\log \det\big(B_{n,\lfloor nt\rfloor}'B_{n,\lfloor nt\rfloor}\big), t \in [0,1]\} and its asymptotic behavior as n\to \infty: a.s. convergence, fluctuations, large deviations. We connect the results for marginals (fixed t) with those obtained by the spectral method.