Gong, Zhengyi2025-02-262025-02-262024-10https://hdl.handle.net/11299/270057University of Minnesota Ph.D. dissertation.October 2024. Major: Mathematics. Advisor: Maury Bramson. 1 computer file (PDF); vii, 157 pages.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.enErgodicityInteracting Particle SystemMajority Vote ProcessMarkov ProcessProbability TheoryThesis or Dissertation