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}=1.7320(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_{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_{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.