Splitting phenomenon in the Sathe-Selberg theorem, mod-Poisson convergence with auxiliary randomisation and universality of the Gamma factor
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913955659644928 |
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| author | Barhoumi-Andréani, Yacine |
| author_facet | Barhoumi-Andréani, Yacine |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_17072 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Splitting phenomenon in the Sathe-Selberg theorem, mod-Poisson convergence with auxiliary randomisation and universality of the Gamma factor Barhoumi-Andréani, Yacine Probability 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. |
| title | Splitting phenomenon in the Sathe-Selberg theorem, mod-Poisson convergence with auxiliary randomisation and universality of the Gamma factor |
| topic | Probability |
| url | https://arxiv.org/abs/2507.17072 |