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

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 value...

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Bibliographic Details
Published in:Europhysics letters 2018-03, Vol.121 (6), p.67004
Main Authors: Hartmann, Alexander K., Le Doussal, Pierre, Majumdar, Satya N., Rosso, Alberto, Schehr, Gregory
Format: Article
Language:English
Subjects:
05.10.Ln
05.20.-y
75.10.Nr
Condensed Matter
Height
High temperature
Importance sampling
Physics
Predictions
Statistical Mechanics
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Summary: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.
ISSN:0295-5075
1286-4854
DOI:10.1209/0295-5075/121/67004