2014/01/01 by Romain Allez, Laure Dumaz
Mathematics · Physics and Astronomy · #BETA (programming language) #Brownian motion #Geometry #Inverse trigonometric functions #Kernel (algebra) #Lévy process #Mathematical analysis #Mathematics #Physics #Point process #Poisson distribution #Poisson point process #Pure mathematics #Random Matrices and Applications #Real line #Scaling #Sine #Statistical physics #Statistics #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #cond-mat.stat-mech #math.PR
paper · pdf · doi:10.1214/ejp.v19-3742
published as Electron. J. Probab. 19 (2014) no. 114, 1-25 · 24 pages, 5 figures
openalex publication_date 2014/01/01 · arxiv created 2014/10/06 · arxiv updated 2014/12/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We study the Sine β process introduced in Valko and Virag, 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 characterised 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 the real line. 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.