Two-dimensional oriented self-avoiding walks with parallel contacts

TL;DR

This paper investigates two-dimensional oriented self-avoiding walks with parallel contacts, revealing a first-order phase transition and analyzing the nature of contacts using enumeration and Monte Carlo methods.

Contribution

It introduces a detailed analysis of phase transitions in OSAWs with parallel contacts, combining enumeration and simulation to uncover new behaviors.

Findings

First-order phase transition at critical beta

Parallel contacts saturate at large walk lengths

Shape of the partition function reconstructed

Abstract

Oriented self-avoiding walks (OSAWs) on a square lattice are studied, with binding energies between steps that are oriented parallel across a face of the lattice. By means of exact enumeration and Monte Carlo simulation, we reconstruct the shape of the partition function and show that this system features a first-order phase transition from a free phase to a tight-spiral phase at $\beta_c=\log(\mu)$, where $\mu =2.638$ is the growth constant for SAWs. With Monte Carlo simulations we show that parallel contacts happen predominantly between a step close to the end of the OSAW and another step nearby; this appears to cause the expected number of parallel contacts to saturate at large lengths of the OSAW.