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

@article{Bothner2012TailDF,
title={Tail decay for the distribution of the endpoint of a directed polymer},
author={Thomas Bothner and Karl Liechty},
journal={arXiv: Mathematical Physics},
year={2012}
}
• Published 16 December 2012
• Mathematics
• arXiv: Mathematical Physics
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 Painlev\'e equation, we prove that as $t\rightarrow\infty$, $\mathbb{P}(|\tcal|>t)=Ce^{-4/3\varphi(t)}t^{-145/32}(1+O(t^{-3/4}))$, where $\varphi(t)=t^3-2t^{3/2}+3t^{3/4}$, and the constant $C… 6 Citations ## Figures from this paper Midpoint Distribution of Directed Polymers in the Stationary Regime: Exact Result Through Linear Response • Physics • 2017 We obtain an exact result for the midpoint probability distribution function (pdf) of the stationary continuum directed polymer, when averaged over the disorder. It is obtained by relating that pdf The endpoint distribution of directed polymers • Mathematics • 2016 Probabilistic models of directed polymers in random environment have received considerable attention in recent years. Much of this attention has focused on integrable models. In this paper, we TASEP on a Ring in Sub-relaxation Time Scale • Physics • 2016 Interacting particle systems in the KPZ universality class on a ring of size L with O(L) number of particles are expected to change from KPZ dynamics to equilibrium dynamics at the so-called Localization and free energy asymptotics in disordered statistical mechanics and random growth models This dissertation develops techniques for analyzing infinite-volume asymptotics in the statistical mechanical setting, and focuses on the low-temperature phases of spin glasses and directed polymers, wherein the ensembles exhibit localization which is physically phenomenological. Distribution function of the endpoint fluctuations of one-dimensional directed polymers in a random potential The explicit expression for the probability distribution function of the endpoint fluctuations of one-dimensional directed polymers in a random potential is derived in terms of the Bethe ansatz Airy processes and variational problems • Mathematics • 2014 We review the Airy processes—their formulation and how they are con- jectured to govern the large time, large distance spatial fluctuations of 1-D random growth models. We also describe formulae ## References SHOWING 1-10 OF 42 REFERENCES Probability distribution of the free energy of the continuum directed random polymer in 1 + 1 dimensions • Mathematics • 2011 We consider the solution of the stochastic heat equation $$\partial_T {\cal Z} = {{1}\over{2}} \partial_X^2 {\cal Z} - {\cal Z} \dot{\cal{W}}$$ with delta function initial condition $${\cal Z} A boundary value problem associated with the second painlevé transcendent and the Korteweg-de Vries equation • Mathematics • 1980 AbstractThe differential equation considered is$$y'' - xy = y|y|^\alpha$$. For general positive α this equation arises in plasma physics, in work of De Boer & Ludford. For α=2, it yields Continuum Statistics of the Airy2 Process • Mathematics • 2013 We develop an exact determinantal formula for the probability that the Airy_2 process is bounded by a function g on a finite interval. As an application, we provide a direct proof that Extremes of N Vicious Walkers for Large N: Application to the Directed Polymer and KPZ Interfaces We compute the joint probability density function (jpdf) PN(M,τM) of the maximum M and its position τM for N non-intersecting Brownian excursions, on the unit time interval, in the large N limit. For Brownian Gibbs property for Airy line ensembles • Mathematics • 2011 Consider a collection of N Brownian bridges$B_{i}:[-N,N] \to \mathbb{R} $, Bi(−N)=Bi(N)=0, 1≤i≤N, conditioned not to intersect. The edge-scaling limit of this system is obtained by taking a weak Scaling for a one-dimensional directed polymer with boundary conditions We study a (1+1)-dimensional directed polymer in a random environment on the integer lattice with log-gamma distributed weights. Among directed polymers, this model is special in the same way as the The Kardar-Parisi-Zhang Equation and Universality Class Brownian motion is a continuum scaling limit for a wide class of random processes, and there has been great success in developing a theory for its properties (such as distribution functions or The right tail exponent of the Tracy–Widom$\beta\$ distribution
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