Exact distribution of the maximal height of p vicious walkers

Abstract

Using path integral techniques, we compute exactly the distribution of the maximal height H_p of p nonintersecting Brownian walkers over a unit time interval in one dimension, both for excursions (p-watermelons with a wall) and bridges (p-watermelons without a wall), for all integer p\ge 1. For large p, we show that < H_p > \sim \sqrt{2p} (excursions) whereas < H_p > \sim \sqrt{p} (bridges). Our exact results prove that previous numerical experiments only measured the pre-asymptotic behaviors and not the correct asymptotic ones. In addition, our method establishes a physical connection between vicious walkers and random matrix theory.

4 pages, 2 figures; minor modifications, added references; published version

References in corpus (4)

• Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems
• Nonintersecting Brownian excursions
• Extreme statistics for time series: Distribution of the maximum relative to the initial value
• Two Bessel Bridges Conditioned Never to Collide, Double Dirichlet Series, and Jacobi Theta Function

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