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