The Ginibre evolution in the large-N limit

TL;DR

This paper investigates the statistical behavior of real eigenvalues in the Ginibre evolution as the matrix size grows large, providing new insights into their correlation functions over time.

Contribution

It introduces a novel analysis of the large-N limit of the Ginibre evolution, including the computation of two-time correlation functions using spin variables.

Findings

Derived the limiting two-time correlation function of spin variables.

Connected the formalism of spin variables to existing fixed-time correlation functions.

Enhanced understanding of eigenvalue dynamics in large random matrices.

Abstract

We analyse statistics of the real eigenvalues of gl(N,R)-valued Brownian motion (the 'Ginibre evolution') in the limit of large $N$. In particular, we calculate the limiting two-time correlation function of spin variables associated with real eigenvalues of the Ginibre evolution. We also show how the formalism of spin variables can be used to compute the fixed time correlation functions of real eigenvalues discovered originally by Forrester and Nagao and Borodin and Sinclair.