Exact Short-Time Height Distribution in the One-Dimensional Kardar-Parisi-Zhang Equation and Edge Fermions at High Temperature
Abstract
We consider the early time regime of the Kardar-Parisi-Zhang (KPZ) equation in 1 +1 dimensions in curved (or droplet) geometry. We show that for short time t , the probability distribution P (H ,t ) of the height H at a given point x takes the scaling form P (H ,t )∼exp [-Φdrop(H )/√{t }] where the rate function Φdrop(H ) is computed exactly for all H . While it is Gaussian in the center, i.e., for small H , the probability distribution function has highly asymmetric non-Gaussian tails that we characterize in detail. This function Φdrop(H ) is surprisingly reminiscent of the large deviation function describing the stationary fluctuations of finite-size models belonging to the KPZ universality class. Thanks to a recently discovered connection between the KPZ equation and free fermions, our results have interesting implications for the fluctuations of the rightmost fermion in a harmonic trap at high temperature and the full counting statistics at the edge.
- Publication:
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Physical Review Letters
- Pub Date:
- August 2016
- DOI:
- arXiv:
- arXiv:1603.03302
- Bibcode:
- 2016PhRvL.117g0403L
- Keywords:
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- Condensed Matter - Statistical Mechanics;
- Condensed Matter - Disordered Systems and Neural Networks;
- Mathematical Physics;
- Mathematics - Probability
- E-Print:
- 5 pages + 7 pages of supplemental material, 3 figures, typos corrected