Le Doussal P, Majumdar SN, Rosso A, Schehr G. Exact Short-Time Height Distribution in the One-Dimensional Kardar-Parisi-Zhang Equation and Edge Fermions at High Temperature.
PHYSICAL REVIEW LETTERS 2016;
117:070403. [PMID:
27563940 DOI:
10.1103/physrevlett.117.070403]
[Citation(s) in RCA: 14] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/25/2016] [Indexed: 06/06/2023]
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[-Φ_{drop}(H)/sqrt[t]] where the rate function Φ_{drop}(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 Φ_{drop}(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.
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