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Exponential moments for disk counting statistics of random normal matrices in the critical regime.(English)Zbl 1528.41081

Summary: We obtain large \(n\) asymptotics for the \(m\)-point moment generating function of the disk counting statistics of the Mittag-Leffler ensemble, where \(n\) is the number of points of the process and \(m\) is arbitrary but fixed. We focus on the critical regime where all disk boundaries are merging at speed \(n^{-\frac{1}{2}}\), either in the bulk or at the edge. As corollaries, we obtain two central limit theorems and precise large \(n\) asymptotics of all joint cumulants (such as the covariance) of the disk counting function. Our results can also be seen as large \(n\) asymptotics for \(n\times n\) determinants with merging planar discontinuities.

MSC:

41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
60B20 Random matrices (probabilistic aspects)
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)

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This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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