From Sine kernel to Poisson statistics

TL;DR

This paper investigates the Sine$_eta$ process as the inverse temperature $eta$ approaches zero, demonstrating its convergence to a Poisson process and revealing a transition from rigid to random eigenvalue distributions.

Contribution

It provides a rigorous proof that the Sine$_eta$ process converges to a Poisson process as $eta$ tends to zero, elucidating the crossover from rigid to random eigenvalue statistics.

Findings

Sine$_eta$ process converges to Poisson process as $eta o 0$

Establishes a smooth transition from rigid to Poisson eigenvalue statistics

Analyzes coupled diffusion processes related to the Brownian carousel

Abstract

We study the Sine$_\beta$ process introduced in [B. Valk\'o and B. Vir\'ag. Invent. math. (2009)] when the inverse temperature $\beta$ tends to 0. This point process has been shown to be the scaling limit of the eigenvalues point process in the bulk of $\beta$-ensembles and its law is characterized in terms of the winding numbers of the Brownian carrousel at different angular speeds. After a careful analysis of this family of coupled diffusion processes, we prove that the Sine$_\beta$ point process converges weakly to a Poisson point process on $\mathbb{R}$. Thus, the Sine$_\beta$ point processes establish a smooth crossover between the rigid clock (or picket fence) process (corresponding to $\beta=\infty$) and the Poisson process.