Disk counting statistics near hard edges of random normal matrices: the multi-component regime.(English)Zbl 1543.60006
The authors consider a two-dimensional point process whose points are separated into two disjoint components by a hard wall, and study the multivariate moment generating function of the corresponding disk counting statistics. They investigate the “hard edge regime” where all disk boundaries are a distance of order \(\frac{1}{n}\) away from the hard wall, where \(n\) is the number of points. It is shown that as \(n\rightarrow +\infty,\) the asymptotics of the moment generating function have the form \(\exp(C_1 n+C_2\ln n+C_3+ \mathfrak{F}_n+\frac{C_4}{\sqrt{n}} + \mathit{O}(n^{-\frac{3}{5}})),\) where the oscillatory term \(\mathfrak{F}_n\) is of order 1 and is given in terms of the Jacobi theta function. Using this theorem, the authors derive various precise results on the disk counting function.
Reviewer: Nasir N. Ganikhodjaev (Tashkent)
MSC:
| 60B20 | Random matrices (probabilistic aspects) |
| 41A60 | Asymptotic approximations, asymptotic expansions (steepest descent, etc.) |
| 60G55 | Point processes (e.g., Poisson, Cox, Hawkes processes) |
Software:
DLMFCite
References:
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