Scaling Analysis for the Adsorption Transition in a Watermelon Network of n Directed Non-Intersecting Walks

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× 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. In honour of the 60th Birthday of R. J. Baxter PACS numbers: 05.50.+q, 05.70.fh, 61.41.+e