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

¹ Institut für Physik, Universität Oldenburg - 26111 Oldenburg, Germany
² LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay - 91405 Orsay, France
³ CNRS-Laboratoire de Physique Théorique de l'Ecole Normale Supérieure - 24 rue Lhomond, 75231 Paris Cedex, France

Received: 7 February 2018
Accepted: 20 April 2018

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 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 also in the region of long times. We present some evidence for the predicted non-trivial crossover in the left tail from the tail exponent to the cubic tail of the Tracy-Widom distribution, although the details of the full scaling form remain beyond reach.