标题： 萨瑟-塞爾伯格定理中的分裂现象，带辅助随机化的模泊松收敛以及Gamma因子的普遍性 显示英文标题

标题： Splitting phenomenon in the Sathe-Selberg theorem, mod-Poisson convergence with auxiliary randomisation and universality of the Gamma factor

摘要： 我们考虑若干随机变量序列，其傅里叶-拉普拉斯变换在适当通过泊松分布随机变量的傅里叶-拉普拉斯变换进行缩放后呈现出相同的\textit{分裂现象}类型（模泊松收敛）。 针对 Kowalski-Nikeghbali 提出的问题，我们解释了普遍项\textit{伽马因子}的出现，这是由于每个模型的共同特征，即存在一个辅助随机化，揭示了一种独立结构。 这个普遍的 Gamma 因子是这种次序随机变量的概率（指数）波动的结果。 属于这一框架的例子包括：随机均匀排列、随机高斯整数、有限域上的随机多项式、有限域值的随机矩阵、随机分拆以及概率数论中的经典 Sathe-Selberg 定理。 对于最后一个例子，触发 Gamma 因子的随机化定义了一个新的概率分布，“delta-Zeta 分布”，它是排列设置中几何分布的类似物。 显示英文摘要

摘要： We consider several sequences of random variables whose Fourier-Laplace transforms present the same type of \textit{splitting phenomenon} when suitably rescaled by the Fourier-Laplace transform of a Poisson-distributed random variable (mod-Poisson convergence). Addressing a question raised by Kowalski-Nikeghbali, we explain the appearance of a universal term, the \textit{Gamma factor}, by a common feature of each model, the existence of an auxiliary randomisation that reveals an independence structure. This universal Gamma factor is a consequence of the probabilistic (exponential) fluctuations of this subordinated random variable. The class of examples that belong to this framework includes: random uniform permutations, random Gauss integers, random polynomials over a finite field, random matrices with values in a finite field, random partitions and the classical Sathe-Selberg theorems in probabilistic number theory. For this last example, the randomisation that triggers the Gamma factor defines a new probability distribution, the ``delta-Zeta distribution'' which is the analogue of the geometric distribution in the permutation setting.