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

Abstract

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

25 pages, 4 figures