Scaling of the atmosphere of self-avoiding walks

TL;DR

This paper investigates the distribution and scaling behavior of atmospheres in self-avoiding walks, providing bounds, conjectures, and Monte Carlo verification, with a focus on trapped walks and their relation to the overall walk set.

Contribution

It introduces a detailed analysis of atmosphere distributions in self-avoiding walks and verifies conjectures through Monte Carlo simulations, highlighting the scaling of trapped walks.

Findings

Trapped walks scale similarly to all self-avoiding walks.

Bounds on the number of walks with fixed atmospheres are established.

Monte Carlo estimates support the conjectures about atmosphere distributions.

Abstract

The number of free sites next to the end of a self-avoiding walk is known as the atmosphere. The average atmosphere can be related to the number of configurations. Here we study the distribution of atmospheres as a function of length and how the number of walks of fixed atmosphere scale. Certain bounds on these numbers can be proved. We use Monte Carlo estimates to verify our conjectures. Of particular interest are walks that have zero atmosphere, which are known as trapped. We demonstrate that these walks scale in the same way as the full set of self-avoiding walks, barring an overall constant factor.