Tails of the endpoint distribution of directed polymers

Quastel, Jeremy; Remenik, Daniel

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 1, pp. 1-17.

Nous prouvons qu’une variable aléatoire $𝒯={arg\; max}_{t\in ℝ}\{{𝒜}_2\left(t\right)-t^2\}$, où ${𝒜}_2$ est un processus ${\mathrm{Airy}}_2$ a une queue qui décroît comme ${\mathrm{e}}^{-c{t}^3}$. La distribution de $𝒯$ 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.

We prove that the random variable $𝒯={arg\; max}_{t\in ℝ}\{{𝒜}_2\left(t\right)-t^2\}$, where ${𝒜}_2$ is the ${\mathrm{Airy}}_2$ process, has tails which decay like ${\mathrm{e}}^{-c{t}^3}$. The distribution of $𝒯$ is a universal distribution which governs the rescaled endpoint of directed polymers in $1+1$ dimensions for large time or temperature.

Classification : 60K35, 82C23
Mots clés : directed random polymers, Kardar–Parisi–Zhang universality class

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