From sine kernel to Poisson statistics

Abstract

We study the Sine β process introduced in [B. Valkó and B. Virág. Invent. math. 177 463-508 (2009)] when the inverse temperature β tends to 0. This point process has been shown to be the scaling limit of the eigenvalues point process in the bulk of β-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 β point process converges weakly to a Poisson point process on R. Thus, the Sine β point processes establish a smooth crossover between the rigid clock (or picket fence) process (corresponding to β = ∞) and the Poisson process. arXiv:1407.5402v2 [math.PR]

Localization of the continuous Anderson Hamiltonian in 1-D

“…In the bulk, the eigenvalues of the β-ensembles converge towards the Sine β process [VV09]. In the small β limit, the Sine β process was also shown to converge towards a (homogeneous) Poisson point process [AD14a] using its characterization via coupled diffusions. Closely related discrete models are Jacobi matrices with random decaying potential [KVV12] and CMV matrices [KS09].…”

Poisson statistics for 1d Schrödinger operators with random decaying potentials

“…Allez-Dumaz [1] showed that Sine β d → P oisson(dλ/2π) as β → 0. This fact can easily be generalized to other processes where the drift term in the corresponding SDE (1.3) is replaced by functions f with mild conditions [12].…”

“…so that, by setting β = n − 1 γ , we can use the idea of [1] : to study the hitting time of Θ nt (λ) to the set πZ, we consider…”

Poisson Statistics for Matrix Ensembles at Large Temperature

“…For real-symmetric ensembles, the same question has recently been considered from a formal point of view. Indeed, in [4,5], Allez and Dumaz considered the β → 0 limit of the Sine-β process and of the β-Tracy-Widom distribution. The β → 0 limit of the Sine-β is also considered by Leblé and Sefaty in [19].…”

“…The β → 0 limit of the Sine-β is also considered by Leblé and Sefaty in [19]. The approach used by [19] is based on approximation theory while [4] use the diffusion representation of the Sine-β process to consider the limit β → 0. One would expect again to prove that when β → 0 simultaneously to N → ∞, the eigenvalues in the vicinity of a point u in the bulk of the spectrum exhibit Poisson statistics.…”