We consider 2-dimensional determinantal processes that are rotationinvariant and study the fluctuations of the number of points in disks. Based on the theory of mod-phi convergence, we obtain Berry–Esseen as well as precise moderate to large deviation estimates for these statistics. These results are consistent with the Coulomb gas heuristic from the physics literature. We also obtain functional limit theorems for the stochastic process $(\# D_r)_{r>0}$ when the radius $r$ of the disk $D_r$ is growing in different regimes. We present several applications to invariant determinantal processes, including the polyanalytic Ginibre ensembles, zeros of the hyperbolic Gaussian analytic function, and other hyperbolic models. As a corollary, we compute the precise asymptotics for the entanglement entropy of (integer) Laughlin states for all Landau levels.

中文翻译：

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不变行列式过程的磁盘计数统计的精确偏差

我们考虑旋转不变的二维行列式过程，并研究磁盘中点数的波动。基于 mod-phi 收敛理论，我们获得了 Berry-Esseen 以及这些统计数据的精确中到大偏差估计。这些结果与物理学文献中的库仑气体启发式一致。当磁盘 $D_r$ 的半径 $r$ 在不同的状态下增长时，我们还获得了随机过程 $(\# D_r)_{r>0}$ 的函数极限定理。我们提出了几种对不变行列式过程的应用，包括多分析 Ginibre 系综、双曲高斯解析函数的零点和其他双曲模型。作为推论，