Exact Short-Time Height Distribution in the One-Dimensional Kardar-Parisi-Zhang Equation and Edge Fermions at High Temperature

Pierre Le Doussal, Satya N. Majumdar, Alberto Rosso, Grégory Schehr
2016 Physical Review Letters  
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) ∼(-Φ_ drop(H)/√(t)) where the rate function Φ_ drop(H) is computed exactly. While it is Gaussian in the center, i.e., for small H, the PDF has highly asymmetric non-Gaussian tails which we characterize in detail. This function Φ_ drop(H) is
more » ... 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 KPZ 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 couting statistics at the edge.
doi:10.1103/physrevlett.117.070403 pmid:27563940 fatcat:ixt3eszfoncyppbhl7vw5h5xxy