2011/05/12 by Steven Delvaux, Delvaux, Steven, Arno B. J. Kuijlaars +5
Mathematics · Physics and Astronomy · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Mathematical functions and polynomials #Probability (math.PR) #Random Matrices and Applications #Spectral Theory in Mathematical Physics #math-ph #math.CA #math.MP #math.PR
paper · pdf · doi:10.48550/arxiv.1105.2481
51 pages, 9 figures
openalex publication_date 2011/05/12 · arxiv created 2011/05/13 · arxiv updated 2011/05/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider a model of n non-intersecting squared Bessel processes with one starting point a>0 at time t=0 and one ending point b>0 at time t=T. After proper scaling, the paths fill out a region in the tx-plane. Depending on the value of the product ab the region may come to the hard edge at 0, or not. We formulate a vector equilibrium problem for this model, which is defined for three measures, with upper constraints on the first and third measures and an external field on the second measure. It is shown that the limiting mean distribution of the paths at time t is given by the second component of the vector that minimizes this vector equilibrium problem. The proof is based on a steepest descent analysis for a 4 × 4 matrix valued Riemann-Hilbert problem which characterizes the correlation kernel of the paths at time t. We also discuss the precise locations of the phase transitions.