Backbone and shortest-path exponents of the two-dimensional $Q$-state Potts model

Abstract

We present a Monte Carlo study of the backbone and the shortest-path exponents of the two-dimensional $Q$-state Potts model in the Fortuin-Kasteleyn bond representation. We first use cluster algorithms to simulate the critical Potts model on the square lattice and obtain the backbone exponents $d_{\mathrm{B}}=1.7320(3)$ and $1.794(2)$ for $Q=2,3$ respectively. However, for large $Q$, the study suffers from serious critical slowing down and slowly converging finite-size corrections. To overcome these difficulties, we consider the O$(n)$ loop model on the honeycomb lattice in the densely packed phase, which is regarded to correspond to the critical Potts model with $Q=n^2$. With a highly efficient cluster algorithm, we determine from domains enclosed by the loops $d_{\mathrm{B}}=1.64339(5),1.73227(8),1.7938(3),1.8384(5),1.8753(6)$ for $Q=1,2,3,2+\sqrt{3},4$, respectively, and $d_{\min }=1.0945(2),1.0675(3),1.0475(3),1.0322(4)$ for $Q=2,3,2+\sqrt{3},4$ respectively. Our estimates significantly improve over the existing results for both $d_{\mathrm{B}}$ and $d_{\min }$. Finally, by studying finite-size corrections in backbone-related quantities, we conjecture an exact formula as a function of $n$ for the leading correction exponent.

14 pages, 9 figures

References in corpus (3)

• A Guide to Stochastic Loewner Evolution and its Applications
• Cluster simulations of loop models on two-dimensional lattices
• Worm Monte Carlo study of the honeycomb-lattice loop model