and asymptotic results

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. 1. Introduction. The problem of vicious walkers was introduced by Fisher [17], who also gave a number of physical applications of the model, such as, for example, to modelling wetting and melting. The general model is one of n random walkers on a d-dimensional lattice who at regular time intervals simultaneously take one step in the direction of one of the allowed lattice vectors such that at no time two walkers occupy the same lattice site. Numerous papers have been written on the subject since then. Most of them analyse the model of vicious walkers in a continuum limit (such as for example [17, 20, 21, 22]). It has been realized only recently that in fact there are many interesting cases in which even exact results in form of nice closed product formulas are available, and that asymptotic analysis can be performed directly on the model, without taking recourse to continuum limits, thus obtaining more precise estimates (see for example [8, 15, 37, 41, 45]).