Large and Moderate Deviations for Entries of Orthogonal Matrices and the Stiefel Manifold

Let AN be distributed according to the Haar probability measure on the orthogonal group O (N) for each N. It is well-known that the upper left mN kN block of NAN with mNkN = o (N) converges in total variation distance to a matrix of same size consisting of i. i. d. standard normal entries as N. In this work, we characterize this convergence on the scale of large deviations. More precisely, we show that under the same condition mNkN = o (N) the empirical measure of entries of this block satisfies a large deviation principle with speed mNkN and rate function given by the relative entropy with respect to the standard normal distribution. Further, we complement the large deviation principle (LDP) obtained by Kabluchko and Prochno in Large deviations for random matrices in the orthogonal group and Stiefel manifold with applications to random projections of product distributions, Annales de l'Institut Henri Poincaré. 60 (2024), 990 -- 1024 for the whole block AN with a moderate deviation principle (MDP). Concretely, we show an MDP for the sequence of matrices βN AN in the product topology, where βN is a sequence of real numbers such that βN = o (N). Here, in contrast to the LDP, the Gaussian behavior of the entries is reflected in the rate function.