Non-Intersecting Squared Bessel Paths and Multiple Orthogonal Polynomials for Modified Bessel Weights

Abstract

We study a model of n non-intersecting squared Bessel processes in the confluent case: all paths start at time t = 0 at the same positive value x = a, remain positive, and are conditioned to end at time t = T at x = 0. In the limit n → ∞, after appropriate rescaling, the paths fill out a region in the tx-plane that we describe explicitly. In particular, the paths initially stay away from the hard edge at x = 0, but at a certain critical time t * the smallest paths hit the hard edge and from then on are stuck to it. For t = t * we obtain the usual scaling limits from random matrix theory, namely the sine, Airy, and Bessel kernels. A key fact is that the positions of the paths at any time t constitute a multiple orthogonal polynomial ensemble, corresponding to a system of two modified Bessel-type weights. As a consequence, there is a 3 × 3 matrix valued Riemann-Hilbert problem characterizing this model, that we analyze in the large n limit using the Deift-Zhou steepest descent method. There are some novel ingredients in the Riemann-Hilbert analysis that are of independent interest.

Non-intersecting squared Bessel paths with one positive starting and ending point

“…To this end, we mainly follow the theme laid out in [38], although the argument is somehow more involved since our RH problem is of size 4 × 4 whereas the dimension treated in [38] is 3 × 3. Furthermore, we have more generic cases to consider here.…”

“…[20,33,36,49]. The case where a > 0 and b = 0 was considered in [38] and [39]. In that situation, all paths, after proper scaling, initially stay away from the hard edge at x = 0, but at a certain critical time t * the lowest paths hit the hard edge and are stuck to it from then on.…”

“…The limiting mean density of the paths at time t is characterized by a non-real solution of an algebraic equation of order three; see also [38,Appendix] and [42] for an interpretation in terms of a vector equilibrium problem with two measures. If t = t * , the local correlations obey the usual scaling limit from the random matrix theory, leading to the sine, Airy, and Bessel kernels [38]. A new kernel was established near the origin at the critical time t * in [39].…”

“…To obtain interesting results, we make a time scaling as in [38] so that the time variable depends on the number of paths n. That is, we replace the variables t and T by 9) so that 0 < t < 1. Now, letting n → ∞, the paths fill out a region in the tx-plane that looks like one of the regions shown in Figures 1 and 2, depending on the product ab.…”

Two Bessel Bridges Conditioned Never to Collide, Double Dirichlet Series, and Jacobi Theta Function

“…nonintersecting random walks) are fixed to the sites located near to the origin. When we impose the condition to stay positive for all vicious walkers, we say "with a wall" (at the origin) [3,8,21,12,20,22]. The height of N-watermelon is the maximum site visited by the vicious walker, who walks the farthest path from the origin.…”