High-precision simulation of the height distribution for the KPZ equation

TL;DR

This paper numerically computes the height distribution of the KPZ equation's solution over a wide range, confirming analytical predictions for rare fluctuations and tail behaviors at different times.

Contribution

It introduces a high-precision numerical method using importance sampling to analyze the KPZ height distribution across large deviations and times.

Findings

Excellent agreement with analytical predictions at short times

Tail exponents 5/2 and 3/2 are preserved over time

Evidence of crossover from large deviation tail to Tracy-Widom distribution

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. Both short and long times are investigated and compared with recent analytical predictions for the large-deviation forms of the probability of rare fluctuations. At short times the agreement with the analytical expression is spectacular. We observe that the far left and right tails, with exponents 5/2 and 3/2 respectively, are preserved until large time. We present some evidence for the predicted non-trivial crossover in the left tail from the 5/2 tail exponent to the cubic tail of Tracy-Widom, although the details of the full scaling form remains beyond reach.