Exact distribution of the maximal height of watermelons

Gregory Schehr 1, Satya N. Majumdar 2, Alain Comtet 2, Julien Randon-Furling 2

Physical Review Letters 101 (2008) 150601

We study p non intersecting one-dimensional Brownian walks, either excursions (p-watermelons with a wall) or bridges (p-watermelons without wall). We focus on the maximal height H_p of these p-watermelons configurations on the unit time interval. Using path integral techniques associated to corresponding models of free Fermions, we compute exactly the distribution of H_p for generic integer p. For large p, one obtains < H_p > \sim \sqrt{2p} for p-watermelons with a wall whereas < H_p > \sim \sqrt{p} for p-watermelons without wall. We point out and solve a discrepancy between these exact asymptotic behaviors and numerical experiments, which recently attracted much attention, and we show that only the pre-asymptotic behaviors of these averages were actually measured. In addition, our method, using tools of many-body physics, provides a simpler physical derivation of the connection between vicious walkers and random matrix theory.

  • 1. Laboratoire de Physique Théorique d'Orsay (LPT),
    CNRS : UMR8627 – Université Paris XI - Paris Sud
  • 2. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS),
    CNRS : UMR8626 – Université Paris XI - Paris Sud