Maximum relative height of one-dimensional interfaces : from Rayleigh to Airy distribution

TL;DR

This paper introduces a new way to define the relative height of one-dimensional fluctuating interfaces, computes the distribution of their maximum height for finite systems, and shows it interpolates between Rayleigh and Airy distributions as a parameter varies.

Contribution

It provides an exact computation of the maximum height distribution for a new relative height definition, connecting Rayleigh and Airy distributions through a parameter.

Findings

Distribution scales as L^{-1/2} with system size L.

Interpolates between Rayleigh and Airy distributions.

Related to the area under a Brownian excursion.

Abstract

We introduce an alternative definition of the relative height h^\kappa(x) of a one-dimensional fluctuating interface indexed by a continuously varying real paramater 0 \leq \kappa \leq 1. It interpolates between the height relative to the initial value (i.e. in x=0) when \kappa = 0 and the height relative to the spatially averaged height for \kappa = 1. We compute exactly the distribution P^\kappa(h_m,L) of the maximum h_m of these relative heights for systems of finite size L and periodic boundary conditions. One finds that it takes the scaling form P^\kappa(h_m,L) = L^{-1/2} f^\kappa (h_m L^{-1/2}) where the scaling function f^\kappa(x) interpolates between the Rayleigh distribution for \kappa=0 and the Airy distribution for \kappa=1, the latter being the probability distribution of the area under a Brownian excursion over the unit interval. For arbitrary \kappa, one finds that it is related to, albeit different from, the distribution of the area restricted to the interval [0, \kappa] under a Brownian excursion over the unit interval.