  We study the performance of Markov chains for the $q$-state ferromagnetic
Potts model on random regular graphs. It is conjectured that their performance
is dictated by metastability phenomena, i.e., the presence of "phases"
(clusters) in the sample space where Markov chains with local update rules,
such as the Glauber dynamics, are bound to take exponential time to escape. The
phases that are believed to drive these metastability phenomena in the case of
the Potts model emerge as local, rather than global, maxima of the so-called
Bethe functional, and previous approaches of analysing these phases based on
optimisation arguments fall short of the task.
  Our first contribution is to detail the emergence of the metastable phases
for the $q$-state Potts model on the $d$-regular random graph for all integers
$q,d\geq 3$, and establish that for an interval of temperatures, which is
delineated by the uniqueness and a broadcasting threshold on the $d$-regular
tree, the two phases coexist. The proofs are based on a conceptual connection
between spatial properties and the structure of the Potts distribution on the
random regular graph, rather than complicated moment calculations.
  Based on this new structural understanding of the model, we obtain various
algorithmic consequences. We first complement recent fast mixing results for
Glauber dynamics by Blanca and Gheissari below the uniqueness threshold,
showing an exponential lower bound on the mixing time above the uniqueness
threshold. Then, we obtain tight results even for the non-local Swendsen-Wang
chain, where we establish slow mixing/metastability for the whole interval of
temperatures where the chain is conjectured to mix slowly on the random regular
graph. The key is to bound the conductance of the chains using a random graph
"planting" argument combined with delicate bounds on random-graph percolation.
