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

References in corpus (16)

• Universality Classes for Extreme Value Statistics
• Airy Distribution Function: From the Area Under a Brownian Excursion to the Maximal Height of Fluctuating Interfaces
• Exact Maximal Height Distribution of Fluctuating Interfaces
• Brownian excursion area, Wright's constants in graph enumeration, and other Brownian areas
• Maximal Height Scaling of Kinetically Growing Surfaces
• Universal Asymptotic Statistics of Maximal Relative Height in One-dimensional Solid-on-solid Models
• Density of near-extreme events
• Maximal height statistics for 1/f^alpha signals
• Finite-size scaling in extreme statistics
• Statistics of extremal intensities for Gaussian interfaces
• Extreme statistics for time series: Distribution of the maximum relative to the initial value
• The first-passage area for drifted Brownian motion and the moments of the Airy distribution
• Extreme Fluctuations in Small-Worlds with Relaxational Dynamics
• Distribution of extremes in the fluctuations of two-dimensional equilibrium interfaces
• Spatial survival probability for one-dimensional fluctuating interfaces in the steady state
• Extreme value statistics and return intervals in long-range correlated uniform deviates

Cited by in corpus (11)

• Random Convex Hulls and Extreme Value Statistics
• From the area under the Bessel excursion to anomalous diffusion of cold atoms
• Universal fluctuations and extreme statistics of avalanches near the depinning transtition
• Extreme value statistics from the Real Space Renormalization Group: Brownian Motion, Bessel Processes and Continuous Time Random Walks
• Distribution of the time at which N vicious walkers reach their maximal height
• Extremal statistics of curved growing interfaces in 1+1 dimensions
• Maximum of N Independent Brownian Walkers till the First Exit From the Half Space
• Distribution of maximum velocities in avalanches near the depinning transition
• Maximum relative height of elastic interfaces in random media
• Area fluctuations on a subinterval of Brownian excursion
• Area distribution and the average shape of a Lévy bridge