Swendsen-Wang dynamics is a Markov chain for sampling the ferromagnetic Potts model using cluster updates
(Swendsen and Wang 1987). For a 
-state configuration at inverse temperature 
, independently retain each graph
edge whose endpoints have the same color with probability 
, and delete every other graph edge. Give each connected
component of the retained graph an independent uniformly
chosen color from the 
colors. This produces the next configuration.
For fixed integers 
and 
, Blanca et al. (2026) prove that on a uniformly
random 
-regular
graph with 
vertices, the chain mixes
in 
cluster updates with high probability for each fixed
.
Here 
is the uniqueness threshold on the infinite 
-regular tree. The result uses
a dichotomy discovered with ChatGPT assistance.
