Airy distribution function: From the area under a brownian excursion to the maximal height of fluctuating interfaces

The Airy distribution function describes the probability distribution of the area under a Brownian excursion over a unit interval. Surprisingly, this function has appeared in a number of seemingly unrelated problems, mostly in computer science and graph theory. In this paper, we show that this distribution function also appears in a rather well studied physical system, namely the fluctuating interfaces. 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 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> 0 where the function f ( x) is the Airy distribution function. This result is valid for both the Edwards Wilkinson (EW) and the Kardar - Parisi - Zhang interfaces. For the free boundary case, the same scaling holds P(h(m), L)= L-1/2F (h(m)L(-1/2)), but the scaling function F( x) is different from that of the periodic case. We compute this scaling function explicitly for the EW interface and call it the F-Airy distribution function. Numerical simulations are in excellent agreement with our analytical results. Our results provide a rather rare exactly solvable case for the distribution of extremum of a set of strongly correlated random variables. Some of these results were announced in a recent Letter [S. N. Majumdar and A. Comtet, Phys. Rev. Lett. 92: 225501 ( 2004)].