## Abstract

We consider an infinite interface of d > 2 dimensions, governed by the Kardar-Parisi-Zhang (KPZ) equation with a weak Gaussian noise which is delta-correlated in time and has short-range spatial correlations. We study the probability distribution of the interface height H at a point of the substrate, when the interface is initially flat. We show that, in stark contrast with the KPZ equation in d < 2, this distribution approaches a non-equilibrium steady state. The time of relaxation toward this state scales as the diffusion time over the correlation length of the noise. We study the steady-state distribution using the optimal-fluctuation method. The typical, small fluctuations of height are Gaussian. For these fluctuations the activation path of the system coincides with the time-reversed relaxation path, and the variance of can be found from a minimization of the (nonlocal) equilibrium free energy of the interface. In contrast, the tails of are nonequilibrium, non-Gaussian and strongly asymmetric. To determine them we calculate, analytically and numerically, the activation paths of the system, which are different from the time-reversed relaxation paths. We show that the slower-decaying tail of scales as In P(H) ∝ |H|, while the faster-decaying tail scales as In P(H) ∝ |H|^{3}. The slower-decaying tail has important implications for the statistics of directed polymers in random potential.

Original language | American English |
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Article number | 053201 |

Journal | Journal of Statistical Mechanics: Theory and Experiment |

Volume | 2018 |

Issue number | 5 |

DOIs | |

State | Published - 1 May 2018 |

### Bibliographical note

Funding Information:We are very grateful to Joachim Krug for illuminating discussions and to Herbert Spohn for a valuable comment. B M acknowledges support from the Israel Science Foundation (Grant No. 807/16) and from the University of Cologne through the Center of Excellence ‘Quantum Matter and Materials’.

Publisher Copyright:

© 2018 IOP Publishing Ltd and SISSA Medialab srl.

## Keywords

- fluctuation phenomena
- kinetic growth processes
- large deviations in non-equilibrium systems
- macroscopic fluctuation theory