The limiting Kac random polynomial and truncated random orthogonal matrices

Abstract

An exact calculation of the eigenvalue statistics of truncated random Haar distributed realorthogonal matrices has recently been carried out by Khoruzhenko, Sommers andZyczkowski. We further develop this calculation and use it to deduce a Pfaffian form of thecorrelations for the zeros of the limiting Kac random polynomial. This contrasts with theforms known from previous studies of the real zeros (a multidimensional Gaussian integralwith the integrand multiplied by the absolute values of the variables) and the complexzeros (a Hafnian).