On Hadamard powers of random Wishart matrices

A famous result of Horn and Fitzgerald is that the beta-th Hadamard power of any n x n positive semi-definite (p.s.d.) matrix with non-negative entries is p.s.d. for all beta <= n-2 and is not necessarily p.s.d. for beta < n - 2, with beta is not an element of N. In this article, we study this question for random Wishart matrix A(n) := XnXnT, where X-n is n x n matrix with i.i.d. Gaussian entries. It is shown that applying x -> vertical bar x vertical bar(alpha) entrywise to An, the resulting matrix is p.s.d., with high probability, for alpha > 1 and is not p.s.d., with high probability, for alpha < 1. It is also shown that if X-n are left perpendicularn(s)right perpendicular x n matrices, for any s < 1, then the transition of positivity occurs at the exponent alpha = s.