SWENDSEN-WANG IS FASTER THAN SINGLE-BOND DYNAMICS

We prove that the spectral gap of the Swendsen-Wang dynamics for the random-cluster model is larger than the spectral gap of a single-bond dynamics, which updates only a single edge per step. For this we give a representation of the algorithms on the joint (Potts/random-cluster) model. Furthermore we obtain upper and lower bounds on the mixing time of the single-bond dynamics on the discrete d-dimensional torus of side length L at the Potts transition temperature for q large enough that are exponential in Ld-1, complementing a result of Borgs, Chayes, and Tetali [Probab. Theory Related Fields, 152 (2012), pp. 509-557].