The height of watermelons with wall

Abstract

We derive asymptotics for the moments as well as the weak limit of the height distribution of watermelons with p branches with wall. This generalises a famous result of de Bruijn, Knuth and Rice on the average height of planted plane trees, and results by Fulmek and Katori et al. on the expected value, respectively the higher moments, of the height distribution of watermelons with two branches. The asymptotics for the moments depend on the analytic behaviour of certain multidimensional Dirichlet series. In order to obtain this information we prove a reciprocity relation satisfied by the derivatives of one of Jacobi's theta functions, which generalises the well known reciprocity law for Jacobi's theta functions.

23 pages, 2 figures; final version accepted for publication

References in corpus (10)

• Non-intersecting Brownian walkers and Yang-Mills theory on the sphere
• Exact distribution of the maximal height of p vicious walkers
• Vicious walkers, friendly walkers and Young tableaux II: With a wall
• Nonintersecting Brownian excursions
• Maximum distributions of bridges of noncolliding Brownian paths
• Functional central limit theorems for vicious walkers
• Two Bessel Bridges Conditioned Never to Collide, Double Dirichlet Series, and Jacobi Theta Function
• The height of watermelons with wall
• Watermelon configurations with wall interaction: exact and asymptotic results
• Asymptotic behaviour of watermelons

Cited by in corpus (13)

• Top eigenvalue of a random matrix: large deviations and third order phase transition
• Non-intersecting Brownian walkers and Yang-Mills theory on the sphere
• Extremes of N vicious walkers for large N: application to the directed polymer and KPZ interfaces
• Reunion probability of N vicious walkers: typical and large fluctuations for large N
• On the joint distribution of the maximum and its position of the Airy2 process minus a parabola
• Nonintersecting Brownian motions on the half-line and discrete Gaussian orthogonal polynomials
• The height of watermelons with wall
• Non-intersecting Brownian bridges and the Laguerre Orthogonal Ensemble
• Noncolliding Brownian Motion with Drift and Time-Dependent Stieltjes-Wigert Determinantal Point Process
• Painleve II in random matrix theory and related fields
• Determinantal Martingales and Correlations of Noncolliding Random Walks
• Composition schemes: q-enumerations and phase transitions
• Determinantal Martingales and Interacting Particle Systems