Cutoff for the swendsen wang dynamics on the complete graph

We study the speed of convergence of the Swendsen Wang (SW) dynamics for the q-state ferromagnetic Potts model on the n-vertex complete graph, known as the mean-field model. The SW dynamics was introduced as an attractive alternative to the local Glauber dynamics, often offering faster convergence rates to stationarity in a variety of settings. A series of works have characterized the asymptotic behavior of the speed of convergence of the mean-field SW dynamics for all q ≤ 2 and all values of the inverse temperature parameter β < 0. In particular, it is known that when β < q the mixing time of the SW dynamics is ·(log n). We strengthen this result by showing that for all β < q, there exists a constant c(β, q) < 0 such that the mixing time of the SW dynamics is c(β, q) log n + ·(1). This implies that the mean-field SW dynamics exhibits the cutoff phenomenon in this temperature regime, demonstrating that this Markov chain undergoes a sharp transition from "far from stationarity"to "well-mixed"within a narrow ·(1) time window. The presence of cutoff is algorithmically significant, as simulating the chain for fewer steps than its mixing time could lead to highly biased samples.