Local average height distribution of fluctuating interfaces

Abstract

Height fluctuations of growing surfaces can be characterized by the probability distribution of height in a spatial point at a finite time. Recently there has been spectacular progress in the studies of this quantity for the Kardar-Parisi-Zhang (KPZ) equation in $1+1$ dimensions. Here we notice that, at or above a critical dimension, the finite-time one-point height distribution is ill-defined in a broad class of linear surface growth models, unless the model is regularized at small scales. The regularization via a system-dependent small-scale cutoff leads to a partial loss of universality. As a possible alternative, we introduce a \emph{local average height}. For the linear models the probability density of this quantity is well-defined in any dimension. The weak-noise theory (WNT) for these models yields the "optimal path" of the interface conditioned on a non-equilibrium fluctuation of the local average height. As an illustration, we consider the conserved Edwards-Wilkinson (EW) equation, where, without regularization, the finite-time one-point height distribution is ill-defined in all physical dimensions. We also determine the optimal path of the interface in a closely related problem of the finite-time \emph{height-difference} distribution for the non-conserved EW equation in $1+1$ dimension. Finally, we discuss a UV catastrophe in the finite-time one-point distribution of height in the (non-regularized) KPZ equation in $2+1$ dimensions.

12 pages, 3 figures, extended version

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Cited by in corpus (8)

• Optimal paths of non-equilibrium stochastic fields: the Kardar-Parisi-Zhang interface as a test case
• Nonequilibrium Steady State of a Weakly-Driven Kardar-Parisi-Zhang Equation
• The derivative of the Kardar-Parisi-Zhang equation is not in the KPZ universality class
• Convergence analysis of a finite element approximation of minimum action methods
• First-passage dynamics of linear stochastic interface models: weak-noise theory and influence of boundary conditions
• Large deviations of the interface height in the Golubović-Bruinsma model of stochastic growth
• Lattice models of random advection and diffusion and their statistics
• Full Statistics of Regularized Local Energy Density in a Freely Expanding Kipnis-Marchioro-Presutti Gas