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

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