Area fluctuations of a Brownian excursion are described by the Airy distribution, which found applications in different areas of physics, mathematics and computer science. Here we generalize this distribution to describe the area fluctuations on a \emph{subinterval} of a Brownian excursion. In the first version of the problem (Model 1) no additional conditions are imposed. In the second version (Model 2) we study the distribution of the area fluctuations on a subinterval given the excursion area on the whole interval. Both versions admit convenient path-integral formulations. In Model 1 we obtain an explicit expression for the Laplace transform of the area distribution on the subinterval. In both models we focus on large deviations of the area by evaluating the tails of the area distributions, sometimes with account of pre-exponential factors. When conditioning on very large areas in Model 2, we uncover two singularities in the rate function of the subinterval area fraction. They can be interpreted as dynamical phase transitions of second and third order.
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 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 dimensi…
Distribution of the time at which N vicious walkers reach their maximal height
研究了一维单位时间内N个不相交的布朗运动（恶性步行者）的极值统计。利用路径积分技术，我们精确地计算了最大值M和到达最大值的时间τ_M的联合分布。我们特别关注不相交的布朗桥（“西瓜没有墙”）和不相交的布朗游览（“西瓜有墙”）。我们详细讨论了这种恶性步行者模型在西瓜配置和随机增长模型在弯曲的几何形状，一方面和定向聚合物在无序介质（DPRM）与一个自由端点的另一方面之间的关系。我们还检查我们的结果使用戴森的布朗运动的数值模拟和面对他们的多核生长模型（PNG）和离散晶格上的DPRM模型的数值模拟。这里展示的一些结果是在最近的一封信中宣布的[J. Rambeau和G. Schehr，Europhys. Lett. 91，60006（2010）]。
