Tails of the endpoint distribution of directed polymers

Quastel, Jeremy ; Remenik, Daniel

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015), p. 1-17 / Harvested from Numdam

Résumé

Nous prouvons qu’une variable aléatoire $\mathcal {T}=\operatorname{arg\,max}_{t\in \mathbb {R}}\lbrace \mathcal {A}_{2}(t)-t^{2}\rbrace $, où $\mathcal {A}_{2}$ est un processus $\mathrm {Airy}_{2}$ a une queue qui décroît comme $\mathrm {e}^{-ct^{3}}$. La distribution de $\mathcal {T}$ est une distribution universelle qui gouverne la position du point final d’un polymère dirigé en dimension $1+1$ à temps grand ou à grande température.

Abstract

We prove that the random variable $\mathcal {T}=\operatorname{arg\,max}_{t\in \mathbb {R}}\lbrace \mathcal {A}_{2}(t)-t^{2}\rbrace $, where $\mathcal {A}_{2}$ is the $\mathrm {Airy}_{2}$ process, has tails which decay like $\mathrm {e}^{-ct^{3}}$. The distribution of $\mathcal {T}$ is a universal distribution which governs the rescaled endpoint of directed polymers in $1+1$ dimensions for large time or temperature.

Détail

Publié le : 2015-01-01

MR 3300961 | Zbl 06412895 | 1 citation dans Numdam

DOI : https://doi.org/10.1214/12-AIHP525
Classification:  60K35,  82C23

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