The Height and Range of Watermelons without Wall

This short note adds some results in parallel to those given in [1] on the random variable height on the set of watermelons with wall. In this paper we prove weak convergence results for the random variables “height” and “range” on the set of watermelons without wall restriction, as well as asymptotics for the moments of the random variable height. The techniques applied are quite similar to those in [1]. Therefore, we only give rather short proofs of the results stated and refer to [1] for more detailed arguments. We refer to [1] for a more complete set of references concerning watermelon configurations (and the more general vicious walkers model). By definition, a p-watermelon of length 2n is a set of p lattice paths in Z satisfying the following conditions: • the paths consist of steps from the set {(1, 1), (1,−1)} only, • the i-th path starts at position (0, 2i) and ends at (2n, 2i) and • the paths are non-intersecting, that is, at no time any two path share the same lattice point.