The limiting Kac random polynomial and truncated random orthogonal matrices

TL;DR

This paper derives a Pfaffian form for the eigenvalue correlations of truncated Haar orthogonal matrices and applies it to analyze the zeros of the limiting Kac random polynomial, revealing new structural insights.

Contribution

It extends previous eigenvalue statistics calculations to obtain a Pfaffian correlation form for the zeros of the limiting Kac polynomial, contrasting with earlier integral and Hafnian forms.

Findings

Pfaffian correlation structure for zeros

New analytical form for eigenvalue statistics

Insights into the zeros of the limiting Kac polynomial

Abstract

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