Global rigidity and exponential moments for soft and hard edge point processes

Abstract

We establish global rigidity upper bounds for universal determinantal point processes describing edge eigenvalues of random matrices. For this, we first obtain a general result which can be applied to general (not necessarily determinantal) point processes which have a smallest (or largest) point: this allows us to deduce global rigidity upper bounds from the exponential moments of the counting function of the process. Combining our general result with known exponential moment asymptotics for the Airy and Bessel point processes, we improve on the best known upper bounds for the global rigidity of the Airy point process, and we obtain new global rigidity results for the Bessel point process.Secondly, we obtain exponential moment asymptotics for Wright's generalized Bessel process and the Meijer-G process, up to and including the constant term. As a direct consequence, we obtain new results for the expectation and variance of the associated counting functions. Furthermore, by combining these asymptotics with our general rigidity theorem, we obtain new global rigidity upper bounds for these point processes.

The Bessel kernel determinant on large intervals and Birkhoff's ergodic theorem

“…The determination of large gap asymptotics is a classical problem in random matrix theory with a long history. There exist various results on large gap asymptotics in the case of a gap on a single interval (the so‐called “one‐cut regime”), see [25, 30, 32, 33, 43, 53] for the sine process, [2, 24] for the Airy process, [27, 34] for the Bessel process, [14, 20, 21] for the Wright's generalized Bessel and Meijer‐ G point processes, [22] for the Pearcey process, [3, 8–13, 15–18, 23] for thinned‐deformations of these universal point processes, and [31, 50, 52] for the sine‐β, Airy‐β and Bessel‐β point processes. We also refer to [44] and [38] for two overviews.…”