Characteristic polynomials of random truncations: moments, duality and asymptotics

TL;DR

This paper investigates the moments of characteristic polynomials of truncated Haar matrices from classical groups, revealing dualities, explicit formulas, and asymptotic behaviors with applications to limit theorems.

Contribution

It provides explicit formulas for moments, explores dualities between matrix size and moments, and derives asymptotic results for characteristic polynomials of truncated Haar matrices.

Findings

Explicit formulas for moments in terms of hypergeometric functions

Duality between matrix size and moments

Asymptotic expansions and limit theorems for large matrices

Abstract

We study moments of characteristic polynomials of truncated Haar distributed matrices from the three classical compact groups O(N), U(N) and Sp(2N). For finite matrix size we calculate the moments in terms of hypergeometric functions of matrix argument and give explicit integral representations highlighting the duality between the moment and the matrix size as well as the duality between the orthogonal and symplectic cases. Asymptotic expansions in strong and weak non-unitarity regimes are obtained. Using the connection to matrix hypergeometric functions, we establish limit theorems for the log-modulus of the characteristic polynomial evaluated on the unit circle.