Partition function zeros of adsorbing Dyck paths

TL;DR

This paper investigates the zeros of partition functions for a solvable model of polymer adsorption using Dyck paths, revealing their convergence to a limaçon and insights into critical behavior.

Contribution

It provides an exact analysis of partition function zeros for a simplified polymer adsorption model, elucidating their asymptotic distribution and edge-singularity formation.

Findings

Zeros converge to a limaçon in the complex plane

Exact locus of zeros can be precisely calculated

Edge-singularity on the positive real axis is characterized

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 lima\c{c}on in the complex plane.