Vector equilibrium problems arising from a model of non-intersecting squared Bessel paths

PABLO ROMÁN; A. B. J. KUIJLAARS; S. DELVAUX; L. ZHANG

Córdoba

Congreso; IV Congreso Latinoamericano De Matemáticos; 2012

In this work, we consider a model of n non-intersecting squared Bessel processes with parameter α, in the confluent case where all particles start, at time t = 0, at the same positive value x = a, remain positive, and end, at time t = T , at a non-zero value x = b. 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.