Fixing \(\beta \ge 0\) and an integer \(q \ge 2\) , consider the ferromagnetic q-Potts measures \(\mu _n^{\beta ,B}\) on finite graphs \(\textsf{G}_n\) on n vertices, with external field strength \(B \ge 0\) and the corresponding random cluster measures \(\varphi ^{q,\beta ,B}_{n}\) . Suppose that as \(n \rightarrow \infty \) the uniformly sparse graphs \(\textsf{G}_n\) converge locally to an infinite d-regular tree \(\textsf{T}_{d}\) , \(d \ge 3\) . We show that the convergence of the Potts free energy density to its Bethe–Peirles 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 \(\textsf{T}_{d}\) . 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 \(\textsf{G}_n\) converge locally to the q-Potts measure on \(\textsf{T}_{d}\) with a boundary wired at color k.
