Darboux Transformations and Random Point Processes: Fig. 1.

2014/01/31 by Marco Bertola, Mattia Cafasso
Mathematics · Physics and Astronomy · #Algebra over a field #Determinantal point process #Distribution (mathematics) #Mathematical functions and polynomials #Point (geometry) #Point process #Point processes and geometric inequalities #Probability distribution #Probability theory #Random Matrices and Applications #Random variable #math-ph #math.MP #nlin.SI

paper · pdf · doi:10.1093/imrn/rnu122

published as Int Math Res Notices (2015) 2015 (15): 6211-6266 · 40 pages, 1 figure (only!), ver2, grammatical corrections

arxiv created 2014/02/03 · openalex publication_date 2014/07/24 · arxiv updated 2015/10/16 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

In this paper, we describe a general method to derive formulas relating the gap probabilities of some classical determinantal random point processes (Airy, Pearcey, and Hermite) with the gap probability of the same processes with “wanderers”, “inliers”, and “outliers”. In this way, we generalize the Painlevé-like formula found by Baik for the Baik–Ben Arous–Péché distribution to many different cases, both in the one and multi-time setting. The method is not ad hoc and relies upon the notion of Schlesinger transformations for Riemann–Hilbert problems.

Citations

• On the distribution of the length of the longest increasing subsequence of random permutations
• Universality of the Local Spacing Distribution¶in Certain Ensembles of Hermitian Wigner Matrices
• Spectral form factor in a random matrix theory
• Non-linear PDEs for gap probabilities in random matrices and KP theory
• Riemann–Hilbert approach to multi-time processes: The Airy and the Pearcey cases
• The Dependence on the Monodromy Data of the Isomonodromic Tau Function
• Fredholm determinants, differential equations and matrix models
• Monodromy perserving deformation of linear ordinary differential equations with rational coefficients. II