An exactly soluble noisy traveling wave equation appearing in the problem of directed polymers in a random medium

TL;DR

This paper provides an exact solution for a noisy traveling wave equation modeling directed polymers in a random medium, revealing how noise influences the wave front and validating a cutoff approximation in the weak noise limit.

Contribution

It introduces an exactly solvable noisy traveling wave model linked to directed polymers, enabling precise analysis of noise effects on wave velocity and diffusion.

Findings

Exact velocity and diffusion constant derived

Validation of cutoff approximation in weak noise limit

Front position influenced mainly by first particle noise

Abstract

We calculate exactly the velocity and diffusion constant of a microscopic stochastic model of $N$ evolving particles which can be described by a noisy traveling wave equation with a noise of order $N^{-1/2}$. Our model can be viewed as the infinite range limit of a directed polymer in random medium with $N$ sites in the transverse direction. Despite some peculiarities of the traveling wave equations in the absence of noise, our exact solution allows us to test the validity of a simple cutoff approximation and to show that, in the weak noise limit, the position of the front can be completely described by the effect of the noise on the first particle.