  Fixing $\beta \ge 0$ and an integer $q \ge 2$, consider the ferromagnetic
$q$-Potts measures $\mu_n^{\beta,B}$ on finite graphs ${\sf G}_n$ on $n$
vertices, with external field strength $B \ge 0$ and the corresponding random
cluster measures $\varphi^{q,\beta,B}_{n}$. Suppose that as $n \to \infty$ the
uniformly sparse graphs ${\sf G}_n$ converge locally to an infinite $d$-regular
tree ${\sf T}_{d}$, $d \ge 3$. We show that the convergence of the Potts free
energy density to its Bethe replica symmetric prediction (which has been proved
in case $d$ is even, or when $B=0$), yields the local weak convergence of
$\varphi^{q,\beta,B}_n$ and $\mu_n^{\beta,B}$ to the corresponding free or
wired random cluster measure, Potts measure, respectively, on ${\sf T}_{d}$.
The choice of free versus wired limit is according to which has the larger
Potts Bethe functional value, with mixtures of these two appearing {as limit
points on} the critical line $\beta_c(q,B)$ where these two values of the Bethe
functional coincide. For $B=0$ and $\beta>\beta_c$, we further establish a
pure-state decomposition by showing that conditionally on the same dominant
color $1 \le k \le q$, the $q$-Potts measures on such edge-expander graphs
${\sf G}_n$ converge locally to the $q$-Potts measure on ${\sf T}_{d}$ with a
boundary wired at color $k$.
