[Sur le couplage d’Arratia et la loi de Dirichlet pour les facteurs d’un entier aléatoire]
Let $x\ge 2$, let $N_x$ be an integer chosen uniformly at random from the set $\mathbb{Z} \cap [1, x]$, and let $(V_1, V_2, \ldots )$ be a Poisson–Dirichlet process of parameter $1$. We prove that there exists a coupling of these two random objects such that
| \[ \mathbb{E}\, \sum _{i \ge 1} \big |{\log P_i- V_i\log x}\big | \asymp 1, \] |
where the implied constants are absolute and $N_x = P_1P_2 \cdots $ is the unique factorization of $N_x$ into primes or ones with the $P_i$’s being non-increasing. This establishes a 2002 conjecture of Arratia, who constructed a coupling for which the left-hand side in the above estimate is $\ll \log \!\log x$, and who also proved that the left-hand side is $\ge 1-o(1)$ for all couplings. In addition, we use our refined coupling to give a probabilistic proof of the Dirichlet law for the average distribution of the integer factorization into $k$ parts proved in 2023 by Leung and we improve on its error term.
Soit $x\ge 2$, soit $N_x$ un entier choisi uniformément au hasard dans l’ensemble $\mathbb{Z} \cap [1, x]$, et soit $(V_1, V_2, \ldots )$ un processus de Poisson-Dirichlet de paramètre $1$. Nous montrons l’existence d’un couplage de ces deux objets aléatoires satisfaisant à
| \[ \mathbb{E}\, \sum _{i \ge 1} \big |{\log P_i- V_i\log x}\big | \asymp 1, \] |
où les constantes implicites sont absolues et où $(P_i)_{i \ge 1}$ est l’unique suite décroissante de nombres premiers ou de uns qui satisfait à $N_x = P_1P_2 \cdots $. Ce résultat établit une conjecture d’Arratia (2002), qui avait auparavant construit un couplage dont l’espérance ci-dessus satisfaisait à $\ll \log \!\log x$, et montré que cette espérance est toujours $\ge 1-o(1)$ pour tout couplage. De plus, nous utilisons le couplage pour fournir une preuve probabiliste de la loi de Dirichlet pour la distribution moyenne des factorisations en $k$ parties d’un entier, résultat initialement établi en 2023 par Leung, et nous en améliorons le terme d’erreur.
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Keywords: Arratia’s coupling, Poisson–Dirichlet process, Wasserstein distance, random integer, prime factorization, divisors, Dirichlet law
Mots-clés : Couplage d’Arratia, processus de Poisson-Dirichlet, distance de Wasserstein, entier aléatoire, factorisation en nombres premiers, diviseurs, loi de Dirichlet
Tony Haddad 1 ; Dimitris Koukoulopoulos 1
CC-BY 4.0
@article{JEP_2025__12__1565_0,
author = {Tony Haddad and Dimitris Koukoulopoulos},
title = {On {Arratia{\textquoteright}s} coupling and the {Dirichlet} law for the factors of a random integer},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {1565--1604},
publisher = {\'Ecole polytechnique},
volume = {12},
year = {2025},
doi = {10.5802/jep.317},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.317/}
}
TY - JOUR AU - Tony Haddad AU - Dimitris Koukoulopoulos TI - On Arratia’s coupling and the Dirichlet law for the factors of a random integer JO - Journal de l’École polytechnique — Mathématiques PY - 2025 SP - 1565 EP - 1604 VL - 12 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.317/ DO - 10.5802/jep.317 LA - en ID - JEP_2025__12__1565_0 ER -
%0 Journal Article %A Tony Haddad %A Dimitris Koukoulopoulos %T On Arratia’s coupling and the Dirichlet law for the factors of a random integer %J Journal de l’École polytechnique — Mathématiques %D 2025 %P 1565-1604 %V 12 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.317/ %R 10.5802/jep.317 %G en %F JEP_2025__12__1565_0
Tony Haddad; Dimitris Koukoulopoulos. On Arratia’s coupling and the Dirichlet law for the factors of a random integer. Journal de l’École polytechnique — Mathématiques, Tome 12 (2025), pp. 1565-1604. doi: 10.5802/jep.317
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