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

Alexander K. Hartmann 1 Pierre Le Doussal 2 Satya N. Majumdar 3 Alberto Rosso 3 Gregory Schehr 3

EPL - Europhysics Letters, European Physical Society/EDP Sciences/Società Italiana di Fisica/IOP Publishing, 2018, 121 (6), 〈10.1209/0295-5075/121/67004〉

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.

  • 1. University of Oldenburg
  • 2. LPTENS - Laboratoire de Physique Théorique de l'ENS
  • 3. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques