On the distribution of the largest real eigenvalue for the real Ginibre ensemble

TL;DR

This paper rigorously analyzes the large deviations of the largest real eigenvalue in the real Ginibre ensemble, confirming Gaussian right tail and exponential left tail behaviors, with implications for particle systems.

Contribution

It provides the first rigorous proofs of the tail behaviors of the largest real eigenvalue distribution in the real Ginibre ensemble, confirming and extending previous heuristic results.

Findings

Right tail of the distribution is Gaussian: $P[\lambda_{max}<t] o 1 - rac{1}{4} ext{erfc}(t)$

Left tail of the distribution is exponential: $P[\lambda_{max}<t] o e^{rac{1}{2\sqrt{2\pi}}\zeta(rac{3}{2})t}$

Results have implications for the distribution of the rightmost particle in annihilating Brownian motions

Abstract

Let $\sqrt{N}+\lambda_{max}$ be the largest real eigenvalue of a random $N\times N$ matrix with independent $N(0,1)$ entries (the `real Ginibre matrix'). We study the large deviations behaviour of the limiting $N\rightarrow \infty$ distribution $P[\lambda_{max}<t]$ of the shifted maximal real eigenvalue $\lambda_{max}$. In particular, we prove that the right tail of this distribution is Gaussian: for $t>0$, \[ P[\lambda_{max}<t]=1-\frac{1}{4}\mbox{erfc}(t)+O\left(e^{-2t^2}\right). \] This is a rigorous confirmation of the corresponding result of Forrester and Nagao. We also prove that the left tail is exponential: for $t<0$, \[ P[\lambda_{max}<t]= e^{\frac{1}{2\sqrt{2\pi}}\zeta\left(\frac{3}{2}\right)t+O(1)}, \] where $\zeta$ is the Riemann zeta-function. Our results have implications for interacting particle systems. The edge scaling limit of the law of real eigenvalues for the real Ginibre ensemble is a rescaling of a fixed time distribution of annihilating Brownian motions (ABM's) with the step initial condition. Therefore, the tail behaviour of the distribution of $X_s^{(max)}$ - the position of the rightmost annihilating particle at fixed time $s>0$ - can be read off from the corresponding answers for $\lambda_{max}$ using $X_s^{(max)}\stackrel{D}{=} \sqrt{4s}\lambda_{max}$.