From Sine kernel to Poisson statistics

Abstract

We study the Sine${}_{\beta }$ process introduced in [B. Valkó and B. Virág. 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.

24 pages, 5 figures

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Cited by in corpus (11)

• Localization of the continuous Anderson Hamiltonian in $1$-d
• The eigenvectors of Gaussian matrices with an external source
• Operator limit of the circular beta ensemble
• Matrix addition and the Dunkl transform at high temperature
• Shape of eigenvectors for the decaying potential model
• The one-dimensional log-gas free energy has a unique minimiser
• Gaussian beta ensembles at high temperature: eigenvalue fluctuations and bulk statistics
• Band-limited mimicry of point processes by point processes supported on a lattice
• The scaling limit of eigenfunctions for 1d random Schrödinger operator
• CLT for real beta-ensembles at high temperature
• Poisson statistics for 1d Schrödinger operators with random decaying potentials