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We consider the problem of sampling from the ferromagnetic Potts andrandom-cluster models on a general family of random graphs via the Glauberdynamics for the random-cluster model. The random-cluster model is parametrizedby an edge probability $p \in (0,1)$ and a cluster weight $q > 0$. We establishthat for every $q\ge 1$, the random-cluster Glauber dynamics mixes in optimal$\Theta(n\log n)$ steps on $n$-vertex random graphs having a prescribed degreesequence with bounded average branching $\gamma$ throughout the fullhigh-temperature uniqueness regime $p<p_u(q,\gamma)$.  The family of random graph models we consider includes the Erd\H{o}s--R\'enyirandom graph $G(n,\gamma/n)$, and so we provide the first polynomial-timesampling algorithm for the ferromagnetic Potts model on Erd\H{o}s--R\'enyirandom graphs for the full tree uniqueness regime. We accompany our resultswith mixing time lower bounds (exponential in the largest degree) for the PottsGlauber dynamics, in the same settings where our $\Theta(n \log n)$ bounds forthe random-cluster Glauber dynamics apply. This reveals a novel and significantcomputational advantage of random-cluster based algorithms for sampling fromthe Potts model at high temperatures.

Sampling from Potts on random graphs of unbounded degree via random-cluster dynamics