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

The partition function for the problem of ii 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 x 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 temperatures, 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 satisfied by the original determinantal solution.