Random sampling of self-avoiding theta-graphs

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.