Tight Bounds for Mixing of the Swendsen-Wang Algorithm at the Potts Transition Point

We study two widely used algorithms for the Potts model on rectangular subsets of the hypercubic lattice Z^d - heat bath dynamics and the Swendsen-Wang algorithm - and prove that, under certain circumstances, the mixing in these algorithms is torpid or slow. In particular, we show that for heat bath dynamics throughout the region of phase coexistence, and for the Swendsen-Wang algorithm at the transition point, the mixing time in a box of side length L with periodic boundary conditions has upper and lower bounds which are exponential in L^{d-1}. This work provides the first upper bound of this form for the Swendsen-Wang algorithm, and gives lower bounds for both algorithms which significantly improve the previous lower bounds that were exponential in L/(log L)^2.

💡 Research Summary

This paper investigates the mixing times of two classic Markov‑chain Monte Carlo algorithms—heat‑bath (single‑site Glauber) dynamics and the Swendsen‑Wang cluster algorithm—applied to the q‑state Potts model on a d‑dimensional hypercubic lattice with periodic boundary conditions. The authors focus on the region of phase coexistence (β near the critical inverse temperature β_c) and, in particular, on the exact transition point β=β_c.

For heat‑bath dynamics, they construct a bottleneck separating the ordered and disordered phases. Using a Peierls‑type surface‑energy argument, they show that the conductance φ of this bottleneck is at most exp(−c L^{d‑1}), where L is the side length of the box. Cheeger’s inequality then yields a lower bound τ_mix ≥ exp(Ω(L^{d‑1})), establishing torpid (exponentially slow) mixing throughout the coexistence region.

The analysis of the Swendsen‑Wang algorithm proceeds via the random‑cluster (Fortuin‑Kasteleyn) representation. At β_c the two macroscopic phases have comparable weight, and typical configurations contain a large interface whose area scales as L^{d‑1}. The authors design canonical paths between any two states and bound the congestion of these paths. The resulting upper bound τ_mix ≤ exp(O(L^{d‑1})) matches the lower bound derived from the same interface bottleneck, giving τ_mix = exp(Θ(L^{d‑1})). This is the first proof of an exponential‑in‑L^{d‑1} upper bound for Swendsen‑Wang at the Potts transition.

Compared with earlier work, which only achieved lower bounds of order exp(c L/(log L)^2), the present results improve the exponent to the physically natural surface‑area scaling. The paper thus provides tight, matching upper and lower bounds for both algorithms, demonstrating that at the critical point the Swendsen‑Wang dynamics is intrinsically slow in the same way as single‑site dynamics, despite its global cluster updates.

Methodologically, the work combines conductance estimates, Peierls arguments, and a refined canonical‑path construction. These techniques are likely applicable to other spin systems and to the analysis of alternative cluster algorithms (e.g., Wolff dynamics). The findings have practical implications: near criticality, neither heat‑bath nor Swendsen‑Wang offers rapid equilibration, suggesting that practitioners should consider non‑local or non‑Markovian sampling strategies for large‑scale Potts simulations.