Truncations of random unitary matrices and Young tableaux

Abstract

Let U be a matrix chosen randomly, with respect to Haar measure, from the unitary group U(d). For any k ≤ d, and any k × k submatrix U k of U, we express the average value of |Tr(Uk)| 2n as a sum over partitions of n with at most k rows whose terms count certain standard and semistandard Young tableaux. We combine our formula with a variant of the Colour-Flavour Transformation of lattice gauge theory to give a combinatorial expansion of an interesting family of unitary matrix integrals. In addition, we give a simple combinatorial derivation of the moments of a single entry of a random unitary matrix, and hence deduce that the rescaled entries converge in moments to standard complex Gaussians. Our main tool is the Weingarten function for the unitary group.