An elector sits on each vertex of an infinite tree of degree k, and has to decide between two alternatives. At each time step, each elector switches to the opinion of the majority of her neighbors. We analyze this majority process when opinions are initialized to independent and identically distributed random variables. In particular, we bound the threshold value of the initial bias such that the process converges to consensus. In order to prove an upper bound, we characterize the process of a single node in the large klimit. This approach is inspired by the theory of mean field spin-glass and can potentially be generalized to a wider class of models. We also derive a lower bound that is non-trivial for small, odd values of k. 1 Definitions and main results Consider a graph G with vertex set V, and edge set E. In the following, we shall denote by ∂i the set of neighbors of i ∈ V, and assume |∂i| < ∞ (i.e. G is locally finite). To each vertex i ∈ V we assign an initial spin σ i (0) ∈ {−1, +1}. The vector of all initial spins is denoted by σ(0). Configuration σ(t) = {σ i (t) : i ∈ V} at subsequent times t = 1, 2, . . . are determined according to the following majority update rule. If ∂i is the set of neighbors of node i ∈ V, we let when j∈∂i σ j (t) = 0. If j∈∂i σ j (t) = 0, then we let with probability 1/2, −σ i (t) with probability 1/2. In order to construct this process, we associate to each vertex i ∈ V, a sequence of i.i.d. Bernoulli(1/2) random variables A i = {A i,0 , A i,1 , A i,2 . . . }, whereby A i,t is used to break the (eventual) tie at time t. A realization of the process is then determined by the triple (G, A, σ(0)), with A = {A i }. In this work we will study the asymptotic dynamic of this process when G is an infinite regular tree of degree k ≥ 2. Let P θ be the law of the majority process where, in the initial configuration, the spins σ i (0)
A discrete-time version of the majority voter model on infinite regular trees was studied in REF .
Majority dynamics on trees and the dynamic cavity method
Annals of Applied Probability 2011, Vol. 21, No. 5, 1694-1748
Mathematics Physics