Scaling analysis for the adsorption transition in a watermelon network of n directed non-intersecting walks

Abstract

The partition function for the problem of n directed non-intersecting walks interacting
via contact potentials with a wall parallel to the direction of the walks has previously been
calculated as an n by n determinant. Here, we describe how to analyse the scaling behaviour
of this problem using alternative representations of the solution. In doing so we derive the
asymptotics of the partition function of a watermelon network of n such walks for all tem-
peratures, and so calculate the associated network exponents in the three regimes: desorbed,
adsorbed, and at the adsorption transition. Furthermore, we derive the full scaling function
around the adsorption transition for all n. At the adsorption transition we also derive a
simple "product form" for the partition function. These results have, in part, been derived
using recurrence relations satised by the original determinantal solution.