Exact Maximal Height Distribution of Fluctuating Interfaces

TL;DR

This paper derives exact distributions for the maximum height of a fluctuating one-dimensional interface in steady state, revealing universal scaling functions related to Brownian excursions, with results confirmed by simulations.

Contribution

It provides the first exact solution for the maximal height distribution of a fluctuating Edwards-Wilkinson interface in one dimension, including different boundary conditions.

Findings

Distribution follows a scaling form with the Airy distribution for periodic boundaries.

Different scaling functions are obtained for free boundary conditions.

Numerical simulations agree with the analytical predictions.

Abstract

We present an exact solution for the distribution P(h_m,L) of the maximal height h_m (measured with respect to the average spatial height) in the steady state of a fluctuating Edwards-Wilkinson interface in a one dimensional system of size L with both periodic and free boundary conditions. For the periodic case, we show that P(h_m,L)=L^{-1/2}f(h_m L^{-1/2}) for all L where the function f(x) is the Airy distribution function that describes the probability density of the area under a Brownian excursion over a unit interval. For the free boundary case, the same scaling holds but the scaling function is different from that of the periodic case. Numerical simulations are in excellent agreement with our analytical results. Our results provide an exactly solvable case for the distribution of extremum of a set of strongly correlated random variables.