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-
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.
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Keywords
rigidity, exponential moments, Muttalib–Borodin ensembles,
product random matrices, random matrix theory, Asymptotic
analysis, large gap probability, Riemann–Hilbert problems