Tail decay for the distribution of the endpoint of a directed polymer

Abstract

We obtain an asymptotic expansion for the tails of the random variable $\tcal=\arg\max_{u\in\mathbb{R}}(\mathcal{A}_2(u)-u^2)$ where ${\mathcal{A}}_2$ is the Airy${}_2$ process. Using the formula of Schehr \cite{Sch} that connects the density function of $\tcal$ to the Hastings-McLeod solution of the second Painlevé equation, we prove that as $t\to \infty$, $\mathbb{P}(|\tcal|>t)=Ce^{-4/3φ(t)}t^{-145/32}(1+O(t^{-3/4}))$, where $\phi (t)=t^3-2t^{3/2}+3t^{3/4}$, and the constant $C$ is given explicitly.

24 pages, 2 figures

References in corpus (7)

• Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions
• Non-intersecting Brownian walkers and Yang-Mills theory on the sphere
• Asymptotics of Tracy-Widom distributions and the total integral of a Painlevé II function
• A determinantal formula for the GOE Tracy-Widom distribution
• On the joint distribution of the maximum and its position of the Airy2 process minus a parabola
• Total integrals of global solutions to Painleve II
• Distribution function of the endpoint fluctuations of one-dimensional directed polymers in a random potential

Cited by in corpus (5)

• Airy processes and variational problems
• TASEP on a ring in sub-relaxation time scale
• Multiplicative functionals on ensembles of non-intersecting paths
• Midpoint distribution of directed polymers in the stationary regime: exact result through linear response
• Localization and free energy asymptotics in disordered statistical mechanics and random growth models