Exact solution of weighted partially directed walks crossing a square

Abstract

We consider partially directed walks crossing a L × L square weighted according to their length by a fugacity t. The exact solution of this model is computed in three different ways, depending on whether t is less than, equal to or greater than 1. In all cases a complete expression for the dominant asymptotic behaviour of the partition function is calculated. The model admits a dilute to dense phase transition, where for 0 < t 1 the partition function scales exponentially in L 2, and when t = 1 there is an intermediate scaling which is exponential in L log L . As such we provide an exact solution of a model of the dilute to dense polymeric phase transition in two dimensions.