A Capacity Criterion for Four-State Potts Reconstruction on Trees

Abstract

For the ferromagnetic four-state broadcast model with 0\lt \lambda\lt 1, we prove that reconstruction on a bounded-degree deterministic rooted tree occurs exactly when its L3 capacity with edge resistances \lambda^{-2|e|} is positive. This gives an exact criterion without regularity or growth-rate assumptions on the tree, including at the exponential critical boundary.