Tails of the endpoint distribution of directed polymers

TL;DR

This paper proves that the tail distribution of the endpoint of directed polymers in 1+1 dimensions decays like an exponential of a cubic function, revealing a universal behavior for large time or temperature.

Contribution

It establishes the tail decay rate of the endpoint distribution of directed polymers, demonstrating its universality across large scales.

Findings

Endpoint distribution tails decay as e^{-ct^3}

Distribution is universal for large time or temperature

Provides rigorous proof of tail behavior

Abstract

We prove that the random variable $\ct=\argmax_{t\in\rr}\{\aip(t)-t^2\}$ has tails which decay like $e^{-ct^3}$. The distribution of $\ct$ is a universal distribution which governs the rescaled endpoint of directed polymers in 1+1 dimensions for large time or temperature.