Edge distribution of thinned real eigenvalues in the real Ginibre ensemble

TL;DR

This paper derives the limiting distribution of the largest real eigenvalue in a thinned real Ginibre ensemble, revealing integrable structures and expressing the distribution as a combination of Fredholm determinants, with applications to tail behavior analysis.

Contribution

It extends integrable structure results to the thinned real Ginibre ensemble and expresses the distribution as a combination of Fredholm determinants, connecting to inverse scattering theory.

Findings

Distribution expressed as a convex combination of Fredholm determinants

Connection established with Zakharov-Shabat inverse scattering theory

Explicit tail behavior and constant factors computed

Abstract

This paper is concerned with the explicit computation of the limiting distribution function of the largest real eigenvalue in the real Ginibre ensemble when each real eigenvalue has been removed independently with constant likelihood. We show that the recently discovered integrable structures in \cite{BB} generalize from the real Ginibre ensemble to its thinned equivalent. Concretely, we express the aforementioned limiting distribution function as a convex combination of two simple Fredholm determinants and connect the same function to the inverse scattering theory of the Zakharov-Shabat system. As corollaries, we provide a Zakharov-Shabat evaluation of the ensemble's real eigenvalue generating function and obtain precise control over the limiting distribution function's tails. The latter part includes the explicit computation of the usually difficult constant factors.