Random sampling of self-avoiding theta-graphs

TL;DR

This paper studies the enumeration and properties of theta-graphs embedded in lattices using advanced Monte Carlo methods, providing insights into their critical exponents and arm-length distributions.

Contribution

It introduces a novel combination of Monte Carlo algorithms to estimate critical exponents for theta-graphs in lattice embeddings.

Findings

Estimated critical exponents for theta-graphs in square and cubic lattices.

Compared exponents for cubic lattice theta-graphs with prime knots.

Provided evidence supporting a conjecture on monodisperse theta-graphs in two dimensions.

Abstract

Theta-graphs are a type of spatial graph with two vertices connected by three edges. We investigate embeddings of theta-graphs in the square and simple cubic lattices, using a combination of the Wang-Landau Monte Carlo method with a variant of the BFACF algorithm which accommodates vertices of degree 3. This allows us to estimate the critical exponents governing the number of theta-graphs and the distributions of the different arm-lengths. For the cubic lattice these values can be compared to the corresponding exponents for prime knots. We also study the number of `monodisperse' theta-graphs where the three arms have the same lengths, and find evidence supporting a conjecture for the critical exponent in two dimensions.