Local average height distribution of fluctuating interfaces

TL;DR

This paper introduces the local average height as a well-defined measure for fluctuating interfaces in linear growth models across dimensions, addressing issues of ill-defined distributions and universality loss in traditional height measurements.

Contribution

It proposes the local average height to overcome ill-defined distributions in high dimensions and develops the weak-noise theory for these models, providing new insights into interface fluctuations.

Findings

Local average height distribution is well-defined in any dimension.

Weak-noise theory yields the optimal interface path conditioned on fluctuations.

Regularization affects universality in height distribution analysis.

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.