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

This paper concentrates on asymptotic properties of determinants of some random symmetric matrices. If Bn,r is a n × r rectangular matrix and B′ n,r its transpose, we study det(B′ n,rBn,r) when n, r tends to infinity with r/n → c ∈ (0, 1). The r column vectors of Bn,r are chosen independently, with common distribution νn. The Wishart ensemble corresponds to νn = N (0, In), the standard normal distribution. We call uniform Gram ensemble the ensemble corresponding to νn = σn, the uniform distribution on the unit sphere Sn−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 { 1 n log det ( B′ n,⌊nt⌋Bn,⌊nt⌋ ) , t ∈ [0, 1]} and its asymptotic behavior as n → ∞: a.s. convergence, fluctuations, large deviations. We connect the results for marginals (fixed t) with those obtained by the spectral method.