Exact height distribution in one-dimensional Edwards-Wilkinson interface with diffusing diffusivity

TL;DR

This paper analytically derives the height distribution of a one-dimensional Edwards-Wilkinson interface with time-dependent diffusivity, revealing a non-Gaussian tail and a scaling behavior of height fluctuations.

Contribution

It provides the first exact analytical computation of the height distribution in an Edwards-Wilkinson interface with stochastic diffusivity, uncovering non-Gaussian features and robustness of exponential tails.

Findings

Height scales as t^{3/4} at a fixed point.

Distribution of scaled height approaches a nonzero constant quadratically at zero.

Exponential decay of the tail for large heights is robust across models with dynamical exponent z>1.

Abstract

We study the height distribution of a one-dimensional Edwards-Wilkinson interface in the presence of a stochastic diffusivity $D(t)=B^2(t)$, where $B(t)$ represents a one-dimensional Brownian motion at time $t$. The height distribution at a fixed point is space is computed analytically. The typical height $h(x,t)$ at a given point in space is found to scale as $t^{3/4}$ and the distribution $G(H)$ of the scaled height $H=h/t^{3/4}$ is symmetric but with a nontrivial shape: while it approaches a nonzero constant quadratically as $H\to 0$, it has a non-Gaussian tail that decays exponentially for large $H$. We show that this exponential tail is rather robust and holds for a whole family of linear interface models parametrized by a dynamical exponent $z>1$, with $z=2$ corresponding to the Edwards-Wilkinson model.