Finite-size scaling functions for directed polymers confined between attracting walls

The exact solution of directed self-avoiding walks confined to a slit of finite width and interacting with the walls of the slit via an attractive potential has been calculated recently. The walks can be considered to model the polymer-induced steric stabilisation and sensitised floculation of colloidal dispersions. The large width asymptotics led to a phase diagram different to that of a polymer attached to, and attracted to, a single wall. The question that arises is: can one interpolate between the single wall and two wall cases? In this paper we calculate the exact scaling functions for the partition function by considering the two variable asymptotics of the partition function for simultaneous large length and large width. Consequently, we find the scaling functions for the force induced by the polymer on the walls. We find that these scaling functions are given by elliptic θ-functions. In some parts of the phase diagram there is more a complex crossover between the single wall and two wall cases and we elucidate how this happens.