Majority vote process on general sructures

Abstract

The majority vote process (MVP) was one of the earliest interacting particle systems (IPS) to be explored, but little progress has been made except on Z and the d-ary regular trees Td, with d ≥ 5. This thesis considers the majority vote process defined on more general structures (S, N ), where S denotes a set, and each N (x) ⊆ S denotes a neighborhood of x. Each site x ∈ S, at each time t ≥ 0, is assumed to have one of two possible opinions, 0 or 1. At rate ε0 (ε1), the opinion of x is updated to 0 (1 irrespective of the opinions in N (x). At rate 1 − ε0 − ε1, the opinion of x aligns with the absolute majority opinion in N (x), which is referred to as the majority dynamics. We investigate the ergodicity and equilibria behavior of the MVP, focusing on their dependence on (S, N ), ε0 and ε1. A central concept will be that of uniformly self-supporting structures. A finite A ⊆ is a self-supporting set if A consisting of all 0s or all 1s implies no site from A can change opinion under the majority dynamics. A structure (S, N ) is a uniformly self-supporting structure if there exists a constant R and a family of self-supporting sets E = {Aα}α∈Λ that covers S, each of which has a diameter bounded by R, such that, for every A ∈ E and x /∈ A, there exists some A′ ∈ E , for which x ∈ A′ and A ∩ A′ = ∅.

Our main results are Theorems 1, 2, 3, and 4. Theorem 1 shows that the majority vote process on a uniformly self-supporting structure (S, N ) is ergodic if one also assumes that either ε0 or ε1 is significantly smaller than the other. Theorem 2 shows that the MVP converges to its unique equilibrium exponentially quickly when additional space homogeneity is assumed on (S, N ). Theorems 3 and 4 consider examples of (S, N ) that are not uniformly self-supporting.

Theorem 3 considers the unoriented majority vote process with nearest-neighbor interactions (UMVP) on Td, d = 3, 4. We demonstrate that when d = 3, the UMVP with sufficiently small noise admits at least two mutually singular equilibria; when d = 4, the UMVP with sufficiently small noise admits uncountably many mutually singular equi- libria. Theorem 4 considers the oriented majority vote process with nearest-neighbor interactions (OMVP) on Zd, d ≥ 4. We demonstrate such an OMVP with sufficiently small noise admits uncountably many mutually singular equilibria.