On the number of zeros of linear combinations of independent characteristic polynomials of random unitary matrices

Abstract

We show that for any linear combination of characteristic polynomials of independent random unitary matrices with the same determinant, the expected proportion of zeros lying on the unit circle tends to 1 as the dimension of the matrices tends to infinity. This result is the random matrix analog of an earlier result by Bombieri and Hejhal on the distribution of zeros of linear combinations of L-functions, and thus is consistent with the conjectured links between the value distribution of the characteristic polynomial of random unitary matrices and the value distribution of L-functions on the critical line.