The generating function for the Bessel point process and a system of coupled Painlevé V equations

Abstract

We study the joint probability generating function for k occupancy numbers on disjoint intervals in the Bessel point process. This generating function can be expressed as a Fredholm determinant. We obtain an expression for it in terms of a system of coupled Painlevé V equations, which are derived from a Lax pair of a Riemann-Hilbert problem. This generalizes a result of Tracy and Widom [24], which corresponds to the case k = 1. We also provide some examples and applications. In particular, several relevant quantities can be expressed in terms of the generating function, like the gap probability on a union of disjoint bounded intervals, the gap between the two smallest particles, and large n asymptotics for n × n Hankel determinants with a Laguerre weight possessing several jumps discontinuities near the hard edge. 2(x − y), α > −1, (1.1)where J α stands for the Bessel function of the first kind of order α (see [22, formula 10.2.2] for a definition of J α ).

Large gap asymptotics for the generating function of the sine point process

“…They expressed F α (x 1 , s 1 ) in terms of the solution of a Painlevé V equation. An analogous result was recently obtained in [5] for an arbitrary m ≥ 1, where it is shown that F α ( x, s) can be expressed identically in terms of a solution to a system of m coupled Painlevé V equations.…”

“…In Section 3, we will take advantage of the fact that has constant jumps to simplify the differential identity using a Lax pair. In the same spirit, other RH problems with constant jumps related to the Airy and Bessel processes have also been used in [18, 19] to simplify the analysis. However, we mention that if our RH problem for reduces to the RH problem considered by Bothner et al .…”

“…In Section 3, we will take advantage of the fact that Φ has constant jumps to simplify the differential identity using a Lax pair. In the same spirit, other RH problems with constant jumps related to the Airy and Bessel processes have also been used in [18,19] to simplify the analysis. However, we mention that if m = 1 our RH problem for Y reduces to the RH problem considered by Bothner et al in [8,9], and that their approach is different from ours and does not rely on the steps 2 and 3 above; instead they have successfully performed a Deift-Zhou steepest descent analysis directly on Y (though in a different regime of the parameters than in this paper).…”

“…Several other probabilistic quantities in the Bessel point process can be expressed in terms of F α , see e.g. [5].…”

Gap Probability at the Hard Edge for Random Matrix Ensembles with Pole Singularities in the Potential

“…More precisely, the coupled Painlevé II systems have been related to the generating function for the Airy point process in [15], to the Fredholm determinants of the Painlevé II and the Painlevé XXXIV kernel in [42]. A coupled Painlevé V system has been related to the generating function for the Bessel point process in [12].…”