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

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(B) = 1.732 0(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(B) = 1.643 39(5), 1.73227(8), 1.793 8(3), 1.838 4(5), 1.875 3(6) for Q = 1, 2, 3, 2 + root 3, 4 respectively, and d(min) = 1.094 5(2), 1.067 5(3), 1.047 5(3), 1.032 2(4) for Q = 2, 3, 2 + root 3, 4, respectively. Our estimates significantly improve over the existing results for both d(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.

Automated summary

We studied the backbone and shortest-path exponents of the two-dimensional Potts model using Monte Carlo simulations. Our results improve upon existing estimates and provide a more accurate understanding of the critical behavior of the model. The study also suggests an exact formula for the leading correction exponent.