Diffusion processes and the asymptotic bulk gap probability for the real Ginibre ensemble

TL;DR

This paper investigates the asymptotic behavior of the gap probability for real eigenvalues in the Ginibre ensemble, linking it to particle annihilation and coalescence processes, and provides both theoretical and numerical insights.

Contribution

It establishes a connection between the real Ginibre ensemble's eigenvalue gaps and particle processes, deriving asymptotic formulas and a finite N determinant representation.

Findings

Asymptotic gap probability is exponential with a specific rate involving the Riemann zeta function.

Derived a determinant formula for finite N gap probability.

Numerical computations confirm the asymptotic formulas.

Abstract

It is known that the bulk scaling limit of the real eigenvalues for the real Ginibre ensemble is equal in distribution to the rescaled $t \to \infty$ limit of the annihilation process $A + A \to \emptyset$. Furthermore, deleting each particle at random in the rescaled $t \to \infty$ limit of the coalescence process $A + A \to A$, a process equal in distribution to the annihilation process results. We use these inter-relationships to deduce from the existing literature the asymptotic small and large distance form of the gap probability for the real Ginibre ensemble. In particular, the leading form of the latter is shown to be equal to $\exp(-(\zeta(3/2)/(2 \sqrt{2 \pi}))s)$, where $s$ denotes the gap size and $\zeta(z)$ denotes the Riemann zeta function. It is shown how this can be rigorously established using an asymptotic formula for matrix Fredholm operators. A determinant formula is derived for the gap probability in the finite $N$ case, and this is used to illustrate the asymptotic formulas against numerical computations.