Watermelon configurations with wall interaction: exact and asymptotic results

Abstract

We perform an exact and asymptotic analysis of the model of $n$ vicious walkers interacting with a wall via contact potentials, a model introduced by Brak, Essam and Owczarek. More specifically, we study the partition function of watermelon configurations which start on the wall, but may end at arbitrary height, and their mean number of contacts with the wall. We improve and extend the earlier (partially non-rigorous) results by Brak, Essam and Owczarek, providing new exact results, and more precise and more general asymptotic results, in particular full asymptotic expansions for the partition function and the mean number of contacts. Furthermore, we relate this circle of problems to earlier results in the combinatorial and statistical literature.

AmS-TeX, 41 pages

References in corpus (5)

• Scaling limit of vicious walks and two-matrix model
• Vicious walk with a wall, noncolliding meanders, and chiral and Bogoliubov-deGennes random matrices
• Vicious Random Walkers and a Discretization of Gaussian Random Matrix Ensembles
• Functional central limit theorems for vicious walkers
• Vicious Walkers in a Potential

Cited by in corpus (6)

• Noncolliding Brownian Motion and Determinantal Processes
• Maximum distributions of bridges of noncolliding Brownian paths
• Two Bessel Bridges Conditioned Never to Collide, Double Dirichlet Series, and Jacobi Theta Function
• The height of watermelons with wall
• Symmetries of statistics on lattice paths between two boundaries
• Determinantal Martingales and Correlations of Noncolliding Random Walks