Exact Maximal Height Distribution of Fluctuating Interfaces

Satya Majumdar 1, Alain Comtet 2, 3

Physical Review Letters 92 (2004) 225501

We present an exact solution for the distribution P(h_m,L) of the maximal height h_m (measured with respect to the average spatial height) in the steady state of a fluctuating Edwards-Wilkinson interface in a one dimensional system of size L with both periodic and free boundary conditions. For the periodic case, we show that P(h_m,L)=L^{-1/2}f(h_m L^{-1/2}) for all L where the function f(x) is the Airy distribution function that describes the probability density of the area under a Brownian excursion over a unit interval. For the free boundary case, the same scaling holds but the scaling function is different from that of the periodic case. Numerical simulations are in excellent agreement with our analytical results. Our results provide an exactly solvable case for the distribution of extremum of a set of strongly correlated random variables.

  • 1. Laboratoire de Physique Théorique - IRSAMC (LPT),
    CNRS : UMR5152 – Université Paul Sabatier - Toulouse III
  • 2. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS),
    CNRS : UMR8626 – Université Paris XI - Paris Sud
  • 3. IHP,
    Institut Henri Poincaré