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1.	Ameur, Yacin, et al. (författare) Disk counting statistics near hard edges of random normal matrices: The multi-component regime 2024 Ingår i: Advances in Mathematics. - : Elsevier BV. - 0001-8708 .- 1090-2082. ; 441 Tidskriftsartikel (refereegranskat)abstract We 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. We investigate the “hard edge regime” where all disk boundaries are a distance of order [Formula presented] away from the hard wall, where n is the number of points. We prove that as n→+∞, the asymptotics of the moment generating function are of the form [Formula presented] and we determine the constants C1,…,C4 explicitly. The oscillatory term Fn is of order 1 and is given in terms of the Jacobi theta function. Our theorem allows us to derive various precise results on the disk counting function. For example, we prove that the asymptotic fluctuations of the number of points in one component are of order 1 and are given by an oscillatory discrete Gaussian. Furthermore, the variance of this random variable enjoys asymptotics described by the Weierstrass ℘-function.

Ameur, Yacin, et al. (författare)
Disk counting statistics near hard edges of random normal matrices: The multi-component regime
2024
Ingår i: Advances in Mathematics. - : Elsevier BV. - 0001-8708 .- 1090-2082. ; 441
Tidskriftsartikel (refereegranskat)abstract
- We 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. We investigate the “hard edge regime” where all disk boundaries are a distance of order [Formula presented] away from the hard wall, where n is the number of points. We prove that as n→+∞, the asymptotics of the moment generating function are of the form [Formula presented] and we determine the constants C1,…,C4 explicitly. The oscillatory term Fn is of order 1 and is given in terms of the Jacobi theta function. Our theorem allows us to derive various precise results on the disk counting function. For example, we prove that the asymptotic fluctuations of the number of points in one component are of order 1 and are given by an oscillatory discrete Gaussian. Furthermore, the variance of this random variable enjoys asymptotics described by the Weierstrass ℘-function.

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