Bridges and random truncations of random matrices

Abstract

We continue to study the squared Frobenius norm of a submatrix of a $n \times n$ random unitary matrix. When the choice of the submatrix is deterministic and its size is $[ns] \times [nt]$, we proved in a previous paper that, after centering and without any rescaling, the two-parameter process converges in distribution to a bivariate Brownian bridge. Here, we consider Bernoulli independent choices of rows and columns with respective parameters $s$ and $t$. We prove by subordination that after centering and rescaling by $n^{-1/2}$, the process converges to another Gaussian process.

This paper has the same purpose as arXiv:1302.6539v1 from the second and third-named authors, but the method of proof is drastically different since now we use the results of arXiv:1007.1366v4 and a subordination