{"id":{"repo_id":"umn","oai_identifier":"oai:conservancy.umn.edu:11299/270057"},"canonical_url":"https://search.dev.ndltd.org/etd/umn/oai:conservancy.umn.edu:11299/270057","repository":{"repo_id":"umn","name":"University of Minnesota","base_url":"https://conservancy.umn.edu/server/oai/request"},"display":{"title":"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.","abstract_html":"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.","abstract_has_math":false,"creators":["Gong, Zhengyi"],"institution":null,"degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2024,"date_issued":"2024-10","date_published":"2024-10","updated_at":"2026-07-24T05:20:07Z","subjects":["Ergodicity","Interacting Particle System","Majority Vote Process","Markov Process","Probability Theory"],"languages":["en"],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/11299/270057","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Gong, Zhengyi"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2025-02-26T15:10:39Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2025-02-26T15:10:39Z"]},{"key":"dc:date.issued","label":"Date","values":["2024-10"]},{"key":"dc:type","label":"Dc Type","values":["Thesis or Dissertation"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Ergodicity","Interacting Particle System","Majority Vote Process","Markov Process","Probability Theory"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/11299/270057"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["University of Minnesota Ph.D. dissertation.October 2024. Major: Mathematics. Advisor: Maury Bramson. 1 computer file (PDF); vii, 157 pages."]},{"key":"dc:description.abstract","label":"Abstract","values":["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."]},{"key":"dc:title","label":"Title","values":["Majority vote process on general sructures"]}]}],"canonical_facts":{"dc:creator":["Gong, Zhengyi"],"dc:date.accessioned":["2025-02-26T15:10:39Z"],"dc:date.available":["2025-02-26T15:10:39Z"],"dc:date.issued":["2024-10"],"dc:description":["University of Minnesota Ph.D. dissertation.October 2024. Major: Mathematics. Advisor: Maury Bramson. 1 computer file (PDF); vii, 157 pages."],"dc:description.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."],"dc:identifier.uri":["https://hdl.handle.net/11299/270057"],"dc:language.iso":["en"],"dc:subject":["Ergodicity","Interacting Particle System","Majority Vote Process","Markov Process","Probability Theory"],"dc:title":["Majority vote process on general sructures"],"dc:type":["Thesis or Dissertation"]},"updated_at":"2026-07-24T05:20:07Z"}