What is the probability that a large random matrix has no real eigenvalues?

TL;DR

This paper analyzes the asymptotic probability that a large real Ginibre matrix has no real eigenvalues, revealing a precise exponential decay rate involving the Riemann zeta function.

Contribution

It provides the first rigorous asymptotic formula for the probability of no real eigenvalues in large real Ginibre matrices, including the case of a fixed number of real eigenvalues.

Findings

The probability decays exponentially with rate involving the Riemann zeta function.

The asymptotic formula holds uniformly for sequences of eigenvalues.

The results connect random matrix theory with special functions and number theory.

Abstract

We study the large-$n$ limit of the probability $p_{2n,2k}$ that a random $2n\times 2n$ matrix sampled from the real Ginibre ensemble has $2k$ real eigenvalues. We prove that, $$\lim_{n\rightarrow \infty}\frac {1}{\sqrt{2n}} \log p_{2n,2k}=\lim_{n\rightarrow \infty}\frac {1}{\sqrt{2n}} \log p_{2n,0}= -\frac{1}{\sqrt{2\pi}}\zeta\left(\frac{3}{2}\right),$$ where $\zeta$ is the Riemann zeta-function. Moreover, for any sequence of non-negative integers $(k_n)_{n\geq 1}$, $$\lim_{n\rightarrow \infty}\frac {1}{\sqrt{2n}} \log p_{2n,2k_n}=-\frac{1}{\sqrt{2\pi}}\zeta\left(\frac{3}{2}\right),$$ provided $\lim_{n\rightarrow \infty} \left(n^{-1/2}\log(n)\right) k_{n}=0$.