The Ginibre ensemble of real random matrices and its scaling limits

TL;DR

This paper derives explicit correlation functions for asymmetric real matrix ensembles, particularly the Ginibre ensemble, and analyzes their behavior in large matrix limits at the bulk and edges.

Contribution

It provides a closed-form Pfaffian expression for correlation functions of real asymmetric matrices and applies it to analyze the Ginibre ensemble's scaling limits.

Findings

Correlation functions expressed as Pfaffians.

Bulk and edge scaling limits computed.

Enhanced understanding of real Ginibre ensemble behavior.

Abstract

We give a closed form for the correlation functions of ensembles of a class of asymmetric real matrices in terms of the Pfaffian of an antisymmetric matrix formed from a $2 \times 2$ matrix kernel associated to the ensemble. We apply this result to the real Ginibre ensemble and compute the bulk and edge scaling limits of its correlation functions as the size of the matrices becomes large.