On the critical points of random matrix characteristic polynomials and of the Riemann $\xi$-function

TL;DR

This paper studies the distribution of critical points of large random matrices and the Riemann ξ-function, revealing stronger repulsion phenomena and linking these distributions under certain hypotheses.

Contribution

It introduces a new point process describing critical points and connects it to the Riemann ξ-function's critical points under the Riemann hypothesis.

Findings

Critical points exhibit stronger level repulsion than eigenvalues.

Conditional on the Riemann hypothesis, the same process describes ξ-function critical points.

The probability of multiple critical points in a short interval is comparable to that of eigenvalues.

Abstract

A one-parameter family of point processes describing the distribution of the critical points of the characteristic polynomial of large random Hermitian matrices on the scale of mean spacing is investigated. Conditionally on the Riemann hypothesis and the multiple correlation conjecture, we show that one of these limiting processes also describes the distribution of the critical points of the Riemann $\xi$-function on the critical line. We prove that each of these processes boasts stronger level repulsion than the sine process describing the limiting statistics of the eigenvalues: the probability to find $k$ critical points in a short interval is comparable to the probability to find $k+1$ eigenvalues there. We also prove a similar property for the critical points and zeros of the Riemann $\xi$-function, conditionally on the Riemann hypothesis but not on the multiple correlation conjecture.