Partition function zeros of adsorbing Dyck paths

Abstract

The zeros of the size-$n$ partition functions for a statistical mechanical model can be used to help understand the critical behaviour of the model as $n\to \infty$. Here we use weighted Dyck paths as a simple model of two-dimensional polymer adsorption, and study the behaviour of the partition function zeros, particularly in the thermodynamic limit. The exact solvability of the model allows for a precise calculation of the locus of the zeros and the way in which an edge-singularity on the positive real axis is formed. We also show that in the limit $n\to \infty$ the zeros converge on a limaçon in the complex plane.