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 walk s 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 alte rnative representations of the solution. In doing so we derive the asymptot ics of the partition function of a watermelon network of n such walks for a ll temperatures, and so calculate the associated network exponents in the t hree regimes: desorbed, adsorbed, and at the adsorption transition. Further more, we derive the full scaling function around the adsorption transition for all n. At the adsorption transition we also derive a simple "product fo rm" for the partition function. These results have, in part, been derived u sing recurrence relations satisfied by the original determinantal solution.