Probing the large deviations of the Kardar-Parisi-Zhang equation at short time with an importance sampling of directed polymers in random media

TL;DR

This paper numerically investigates the short-time large deviations of the KPZ equation's height distribution by mapping it to directed polymers, achieving highly precise tail probabilities and confirming recent analytical predictions.

Contribution

It introduces an importance sampling method to accurately compute the KPZ height distribution tails at short times, validating analytical large-deviation forms.

Findings

Excellent agreement with analytical large-deviation predictions.

Accurate tail probabilities down to 10^{-1000}.

Studied different initial conditions including flat, stationary, and droplet.

Abstract

The one-point distribution of the height for the continuum Kardar-Parisi-Zhang (KPZ) equation is determined numerically using the mapping to the directed polymer in a random potential at high temperature. Using an importance sampling approach, the distribution is obtained over a large range of values, down to a probability density as small as $10^{-1000}$ in the tails. The short time behavior is investigated and compared with recent analytical predictions for the large-deviation forms of the probability of rare fluctuations, showing a spectacular agreement with the analytical expressions. The flat and stationary initial conditions are studied in the full space, together with the droplet initial condition in the half-space.