Airy Distribution Function: From the Area Under a Brownian Excursion to the Maximal Height of Fluctuating Interfaces

By Satya N. Majumdar, A. Comtet. 2004

160 citations recorded by the source corpus.

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(hm,L) of the maximal height hm (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(hm,L)=L−1/2f(hmL−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(hm,L)=L−1/2F(hmL−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)].

Subfields: Condensed Matter Physics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/c96c791b7bdf4da5220396f88a78bc5d99a71242

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Citations in the corpus

Extreme value statistics of correlated random variables: A pedagogical review (2019, methodology): “For such weakly correlated variables one can construct a heuristic argument to study the extreme statistics [24], as we describe now.”
Brownian excursion area, Wright’s constants in graph enumeration, and other Brownian areas (2007, methodology): “Majumdar and Comtet [35] give the formula (using C(αj) for our r 2 j ), where rj = AI(aj)/Ai (aj) as in Section 22, ψdm(t) = 2 −1/3t2/3 ∞”
Nonintersecting Brownian Excursions (2006, background): “) Both the distribution of A and the moments of A are known (the moments are known recursively); and perhaps surprisingly, they have found numerous applications [10, 19] outside their original setting.”