AbstractWe establish rapid mixing of the random-cluster Glauber dynamics on random $$\varDelta $$
Δ
-regular graphs for all $$q\ge 1$$
q
≥
1
and $$p<p_u(q,\varDelta )$$
p
<
p
u
(
q
,
Δ
)
, where the threshold $$p_u(q,\varDelta )$$
p
u
(
q
,
Δ
)
corresponds to a uniqueness/non-uniqueness phase transition for the random-cluster model on the (infinite) $$\varDelta $$
Δ
-regular tree. It is expected that this threshold is sharp, and for $$q>2$$
q
>
2
the Glauber dynamics on random $$\varDelta $$
Δ
-regular graphs undergoes an exponential slowdown at $$p_u(q,\varDelta )$$
p
u
(
q
,
Δ
)
. More precisely, we show that for every $$q\ge 1$$
q
≥
1
, $$\varDelta \ge 3$$
Δ
≥
3
, and $$p<p_u(q,\varDelta )$$
p
<
p
u
(
q
,
Δ
)
, with probability $$1-o(1)$$
1
-
o
(
1
)
over the choice of a random $$\varDelta $$
Δ
-regular graph on n vertices, the Glauber dynamics for the random-cluster model has $$\varTheta (n \log n)$$
Θ
(
n
log
n
)
mixing time. As a corollary, we deduce fast mixing of the Swendsen–Wang dynamics for the Potts model on random $$\varDelta $$
Δ
-regular graphs for every $$q\ge 2$$
q
≥
2
, in the tree uniqueness region. Our proof relies on a sharp bound on the “shattering time”, i.e., the number of steps required to break up any configuration into $$O(\log n)$$
O
(
log
n
)
sized clusters. This is established by analyzing a delicate and novel iterative scheme to simultaneously reveal the underlying random graph with clusters of the Glauber dynamics configuration on it, at a given time.
Glauber dynamics for Ising models on random
regular graphs: cut-off and metastability
