The broadcast voter model: Stationary measures
The classical voter model is dual to coalescing random walks, in much the same way that the Kingman coalescent describes genealogies through binary mergers. Motivated by the passage from the Kingman to the~\(Λ\)-coalescent, where multiple mergers are allowed, we introduce the~\(Λ\)-broadcast voter model, in which a voter may transmit its opinion simultaneously to several neighbors according to a mechanism governed by a finite measure~\(Λ\) on~\([0,1]\); the classical voter model is recovered when~\(Λ=δ_0\). We show that our model is well-defined on general graphs with bounded degrees, and we construct a family of stationary measures~\({μ_{α,Λ}}\), parametrized by bounded harmonic functions, that characterize the extremal stationary measures according to the finite or infinite collision property of the underlying random walk. We then study the dependence of the stationary measures on the broadcasting measure $Λ$ through a finite number of its moments. We give general conditions under which different moment profiles yield different stationary measures and obtain, in particular, a complete characterization on trees with bounded degrees and on nearest-neighbour~\(\mathbb Z^d\),~\(d\ge3\). We also prove continuity of the stationary measures with respect to the moment profile. Our main tool is a dual system of coordinated coalescing random walks, whose individual particles are simple random walks but which may jump and coalesce simultaneously; a coupling with independent random walks allows us to relate their long-time behaviour to ordinary random walk collision properties.