论文标题
双曲线高斯分析功能的零的拥挤
Overcrowding for zeros of Hyperbolic Gaussian analytic functions
论文作者
论文摘要
我们考虑了单位磁盘中高斯分析函数的家族$ \ {f_l \} _ {l> 0} $，其零设置相对于双曲线异构体的零设置而区别。令$ n_l \ left（r \右）$为半径$ r $ $ f_l $的零数。我们研究了罕见事件的渐近概率，在该事件中，零为$ r \ uparrow1 $，即，对于每个$ l> 0 $，我们正在寻找概率$ \ mathbb {p} \ mathbb {p} \ weft [n_l（r）\ geq v v with $ y的差异（p} p} $ \ mathbb {e} \ left [n_l \ left（r \ right）\ right] $。佩雷斯（Peres）和维拉格（Virág）表明，对于$ l = 1 $（并且只有那时），零集构成了确定点过程，从而使许多明确的计算成为可能。奇怪的是，与对平面模型的理解相反，似乎$ l <<1 $，当$ v $接近$ \ mathbb {e} \ left [n_l \ left [n_l \ left（r \ r \ right）\ right] $的可能性比零不足的可能性要小得多。
We consider the family $\{f_L\}_{L>0}$ of Gaussian analytic functions in the unit disk, distinguished by the invariance of their zero set with respect to hyperbolic isometries. Let $n_L\left(r\right)$ be the number of zeros of $f_L$ in a disk of radius $r$. We study the asymptotic probability of the rare event where there is an overcrowding of the zeros as $r\uparrow1$, i.e. for every $L>0$, we are looking for the asymptotics of the probability $\mathbb{P}\left[n_L(r)\geq V(r)\right]$ with $V\left(r\right)$ large compared to the $\mathbb{E}\left[n_L\left(r\right)\right]$. Peres and Virág showed that for $L=1$ (and only then) the zero set forms a determinantal point process, making many explicit computations possible. Curiously, contrary to the much better understood planar model, it appears that for $L<1$ the exponential order of decay of the probability of overcrowding when $V$ is close to $\mathbb{E}\left[n_L\left(r\right)\right]$ is much less than the probability of a deficit of zeros.
