{"id":"be8b38af-8276-4df5-950a-2c902d56ed1a","arxiv_id":"2501.16202","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"For every d≥3 and small enough noise, the noisy majority vote process on the infinite d-regular tree has multiple equilibrium measures.","lead":"This paper proves that the noisy majority vote process on infinite trees with at least three neighbors is non-ergodic for small noise: different starting opinions can settle into different stable opinion distributions. The result extends a 2021 proof that worked only for trees with five or more neighbors, closing the last open case for regular trees.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The step from Theorem 1.2 to Theorem 1.1 is missing a compactness argument: uniform one-point bounds do not by themselves yield distinct invariant measures, and 'follows directly' is not a complete proof of non-ergodicity.","rationale":"We read the paper as a proof of Theorem 1.2 via the induction in Theorem 2.8, with Theorem 1.1 claimed as a direct corollary. The reader's weakest assumption, Lemma 2.15, is the inequality ξ*(A,B)+|Prr(A,B)|≤|B|−r. We spot-checked this on representative configurations (single odd vertices, lines of odd vertices, branched clusters) and found it holds, often with equality; the proof through (4.1) and induction on ξ* is internally sound. The combinatorial enumeration in Lemma 2.22 is elaborate, but the final bound (3.3) follows from the stated injectivity and dimension counts if those hold, and we found no specific error. The genuine load-bearing gap is the passage from Theorem 1.2 to Theorem 1.1: the paper asserts domination and says Theorem 1.1 follows directly, but non-ergodicity requires constructing two distinct invariant measures. A standard Cesàro compactness argument supplies this, but it is omitted. Because this step is standard and short, the appropriate verdict remains conditional pending its addition, which matches the reader's CONDITIONAL verdict. We therefore propose a concrete test: a full write-out of that compactness argument, plus an optional brute-force verification of Lemma 2.15 on small trees.","tokens_in":34474,"tokens_out":36836,"duration_ms":310735,"concrete_test":"Write out the missing compactness argument in full: define P_t as the law of the majority vote process at time t from the all-plus initial condition, set \\bar{P}_T = (1/T)Σ_{t=0}^{T-1}P_t, take a subsequential limit μ in the product topology, and verify (i) μ is invariant using the Feller property and (ii) μ(σ_v=-1)≤c via finite-cluster approximation. If the argument cannot be completed — for instance, if the semigroup is not Feller on the product space or the bound does not pass to the limit — then Theorem 1.1 is not established. As a secondary numerical check, exhaustively verify Lemma 2.15 on small trees (all odd clusters A of size ≤6 and all B⊂N(A) with B a union of non-adjacent even clusters) to confirm that ξ*(A,B)+|Prr(A,B)|≤|B|−r always holds.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's advertised conclusion is non-ergodicity, i.e., existence of multiple equilibrium measures. Theorem 1.2 proves only a uniform bound for the minus biased process, P(σ_v(t)=-1)≤c<0.5 for every vertex v and every time t, and the majority vote process is then dominated by the minus biased process. The introduction claims 'the probability of a vertex being 1 has a limit strictly bigger than 1/2, as t→∞,' but no convergence is established anywhere in the paper. To obtain Theorem 1.1 one must construct two distinct invariant measures. The standard route is to use the Feller property on the compact state space {−1,1}^V: the Cesàro means (1/T)Σ_{t=0}^{T-1}P_t, where P_t is the law from the all-plus configuration, have a subsequential limit, and any such limit is invariant; the bound P(σ_v=-1)≤c passes to the limit, giving an invariant μ_+ with μ_+(σ_v=-1)≤c. By global spin-flip symmetry, the all-minus start gives a limit ar{μ}_+ with ar{μ}_+(σ_v=-1)≥1−c>0.5, so the two are distinct. This compactness argument is standard, but it is not written out, and the sentence 'Theorem 1.1 follows from Theorem 1.2 directly' is therefore an omitted step in the proof of the central claim. This is the most load-bearing gap because it is the exact point where the technical one-point bound is converted into the advertised non-ergodicity conclusion. The reader's weakest assumption, Lemma 2.15, appears correct: the inequality ξ*(A,B)+|Prr(A,B)|≤|B|−r holds in representative small examples, and the inductive proof in Section 4 is coherent. The elaborate enumeration in Lemma 2.22 is harder to check by hand, but we found no concrete flaw there either.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that the noisy majority vote process on the infinite d-regular tree T_d is not ergodic for every d≥3 and sufficiently small noise ε, extending Bramson and Gray's earlier result for d≥5. The main technical result is Theorem 1.2: for the dominated \"minus biased\" process started from all plus, P(σ_v(t)=-1)≤c<1/2 uniformly in v and t. This is obtained through the inductive bound in Theorem 2.8, whose proof is carried by explicit combinatorial estimates and enumeration lemmas (notably Lemmas 2.15, 2.21, 2.22). The paper asserts that Theorem 1.1 follows from Theorem 1.2 by domination, and that a version for arbitrary trees of maximal degree d (Theorem 1.3) follows similarly.","tokens_in":34832,"tokens_out":5963,"duration_ms":59702,"significance":"If correct, the result closes the d=3,4 gap for a natural opinion-formation model on trees and provides a genuinely new proof strategy. The core induction is detailed, self-contained, and does not rely on the authors' prior results or on numerically fitted parameters; the constants c1,c2,c3 are chosen after the estimates are derived, and the central closing inequality (2.19) and the multi-cluster bound (2.45) are explicit. The main weakness is not in the induction itself but in the transition from the one-point bound to the advertised non-ergodicity conclusion, which is currently asserted rather than proved. That transition is a standard compactness argument and is readily fixable.","major_comments":[{"comment":"The assertion that Theorem 1.1 follows from Theorem 1.2 directly is incomplete. Theorem 1.2 is a uniform bound on one-point marginals at finite times, whereas non-ergodicity means the existence of at least two distinct invariant measures. A compactness step is required and should be written out. Since {−1,1}^V is compact and the process is Feller, the Cesàro means (1/T)Σ_{t=0}^{T-1} P(σ(t) ∈ ·) starting from the all-plus configuration have a subsequential limit μ_+; any such limit is invariant, and the bound P(σ_v(t)=-1)≤c passes to the limit, giving μ_+(σ_v=-1)≤c. Starting from the all-minus configuration and using global spin-flip symmetry yields an invariant measure μ_- with μ_-(σ_v=-1)≥1-c>1/2, so μ_+ and μ_- are distinct. Without this argument the advertised conclusion of non-ergodicity is not proved.","section":"Section 1.4, paragraph after Theorem 1.2"}],"minor_comments":[{"comment":"The statement that \"the probability of a vertex being 1 has a limit strictly bigger than 1/2, as t→∞\" is stronger than what is established: Theorem 1.2 gives only a uniform upper bound away from 1/2 for all t, not convergence. Please rephrase this as a bound on liminf or as a uniform-in-time bound, and note that convergence is not needed for Theorem 1.1.","section":"Section 1.3"},{"comment":"The sentence \"the minus biased vote process on T dominates the minus biased vote process on T_d\" appears to have the domination direction reversed: a tree of maximal degree d has no more neighbors at each vertex than T_d, so fewer minus votes are available and the process on T is dominated by, rather than dominates, the process on T_d. The claimed conclusion is still correct, but the wording should be fixed.","section":"Section 1.4, Theorem 1.3"},{"comment":"The sentence \"Our goal is to bound the probability that σ_v(T)=1 for all v∈A\" should read σ_v(T)=-1; the surrounding computation works with the event that A is updated to all minuses.","section":"Section 2.3, first paragraph"},{"comment":"Lemma 2.24 states the bounds for \"any r1,r2≥0\", but the quantity F(r1,r2) is defined by binomial coefficients (m_i−1 choose r_i−1) and is used only for r1+r2≥1. Please state the quantifier as r1+r2≥1 to avoid division by zero in the binomial convention.","section":"Lemma 2.24"},{"comment":"The geometric-series simplification leading to (2.19) requires explicit smallness conditions such as 32(d−1)^2 ε<1 and 16(d−1)^2 ε<1; it would help to state these before the formula rather than only saying \"ε small enough\" afterwards.","section":"Equation (2.19)"}],"recommendation":"major_revision","confidential_remarks":"I see no circularity in the induction and no reason to doubt the main technical estimates, though I did not independently verify every enumeration bound in Section 3. The missing compactness argument for Theorem 1.1 is routine and should be added; after that, the paper is very likely acceptable. The overstatement in Section 1.3 also needs correction. This is a strong paper that merits a major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the paper is a real advance. It extends Bramson–Gray from d≥5 to all d≥3, and the d=3,4 cases are not a routine parameter chase—the domination argument genuinely breaks down there, and the authors build a new induction on update histories, with single/double trifurcation counting lemmas to close it. Theorem 1.2, the uniform one-point bound for the minus biased process, looks well supported: the induction is detailed, the closing bounds in (2.19) and (2.45) are explicit, and the constants are chosen after the inequalities, no fitting.\n\nI mostly agree with the reader. The weakest spot is the step from Theorem 1.2 to Theorem 1.1. Section 1.3 says the probability of a vertex being 1 has a limit strictly bigger than 1/2 as t→∞, but no convergence is proved anywhere. What Theorem 1.2 gives is a uniform upper bound c<0.5 for the minus biased process, and by domination the same for the majority vote process started from all-plus. To get multiple equilibrium measures you need the standard compactness argument: Feller dynamics on {−1,1}^V, Cesàro means from all-plus and all-minus starts, pass the bound through the limit, and use spin-flip symmetry to see the two invariant measures differ. This is routine for the intended audience, but it is not written out, and 'follows directly' is doing some work. A referee should ask for it to be included; it is a small fix, not a structural problem.\n\nThe elaborate enumeration lemmas (2.21, 2.22, and the ψ-map business) are the part I could not fully verify by hand. I did not find a concrete flaw; Lemma 2.15 checks out on examples and the leaf-cutting/order argument is coherent. The burden is on a careful referee to comb through Section 3 slowly.\n\nThe paper is written for people who work on interacting particle systems and phase transitions on trees. It deserves a serious referee: the main result is new, the method is genuinely new, and the proof is detailed enough to check. Recommended: send to peer review, and ask the authors to turn the 'directly' into an explicit compactness argument (or a citation), and to soften the limit claim in Section 1.3 unless they prove it.","headline":"Solid new proof for d=3,4 non-ergodicity on trees; the only real gap is a standard compactness step that the authors should spell out.","tokens_in":35398,"tokens_out":2192,"would_cite":true,"duration_ms":19673,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","82C22"],"pacs":[],"model":"deepseek-v4-flash","headline":"The noisy majority vote process on the infinite d-regular tree is non-ergodic for every d ≥ 3 and small enough noise: starting from all plus, any fixed vertex stays plus with probability uniformly above 1/2.","keywords":["noisy majority vote","non-ergodicity","infinite regular tree","minus biased process","multiple equilibrium measures","interacting particle systems","phase transition"],"falsifier":"Test Lemma 2.15 directly on a small example: take an odd cluster $A$ in $\\mathbb{T}_3$ and enumerate all minimal even sets $B$ with $\\Prr(A,B) \\neq \\emptyset$; any instance with $\\xi^*(A,B)+|\\Prr(A,B)| > |B|-r$ would refute the lemma. Alternatively, for $d=3$ and a small fixed $\\epsilon$, compute the supremum over $t$ of $\\mathbb{P}(\\sigma_v(t)=-1)$ starting from all plus; a value at or above $1/2$ would falsify Theorem 1.2.","tokens_in":34246,"feed_emoji":"🌳","tokens_out":6906,"duration_ms":63603,"temperature":0.7,"pith_summary":"The paper proves that on the infinite d-regular tree with d ≥ 3, the noisy majority vote process is non-ergodic: for small enough noise there are multiple equilibrium measures, not a unique one. Previous work had settled d ≥ 5 and d = 2 (where the equilibrium is unique); this closes d = 3 and 4 and covers all higher degrees. The proof works by comparing the process to a 'minus biased' variant and showing that, starting from all plus, the chance any fixed vertex is minus at any time stays below a constant strictly less than 1/2. That uniform bound is what forces the existence of multiple equilibrium measures.","feed_headline":"Noise loses on tree majority vote for every degree ≥ 3","feed_subtitle":"New proof shows a fixed vertex keeps a plus bias above 1/2, giving multiple equilibria where only d≥5 was known.","key_machinery":"The argument is an induction on time using update histories, in which past minus vertices responsible for a current minus cluster are traced backward. The load-bearing identity is Lemma 2.15: for an odd cluster $A$ and a minimal set $B$ of even vertices that can vote it down, $\\xi^*(A,B) + |\\Prr(A,B)| \\le |B| - r$, where $r$ is the number of connected components of $B$, $\\Prr(A,B)$ is the set of vertices of $A$ with at least two neighbors in $B$, and $\\xi^*$ counts single and double trifurcations with weight $p-2$. This inequality produces the extra $\\epsilon^{\\xi^*}$ factor that lets the induction close in the single-cluster case. Two enumeration lemmas bound the number of possible past configurations by reducing each cluster to its depth-first-search order and recording capacities of black and white components; together these turn the probability estimate into a convergent counting problem.","core_discovery":"The central claim is Theorem 1.1: for every $d \\ge 3$ there is an $\\epsilon_0(d) > 0$ such that for $0 < \\epsilon \\le \\epsilon_0$ the noisy majority vote process on $\\mathbb{T}_d$ is not ergodic. The engine is Theorem 1.2, which shows for the minus biased vote process -- a dominated variant where a vertex turns minus if at least two neighbors are minus or a noise event occurs -- that $\\mathbb{P}(\\sigma_v(t) = -1) \\le c < 0.5$ for every vertex $v$ and every time $t$, with $c$ depending only on $d$. Since the original noisy majority vote process dominates this minus biased process, the same uniform upper bound holds for the original process. The proof is an induction on time over the probability that a collection of vertices is minus, and the induction closes because a key combinatorial inequality supplies an extra factor of $\\epsilon$ small enough to absorb all counting constants.","pith_inferences":["A natural testable extension the authors do not pursue is locating the actual noise threshold $\\epsilon_d$ on $\\mathbb{T}_3$; the proof gives an explicit but extremely small lower bound, and simulation could show the true threshold is far larger.","The tree-specific inequality in Lemma 2.15 should extend to any infinite tree with minimum degree at least 3, even without regularity, which would upgrade Theorem 1.3 into a non-ergodicity statement for all such trees.","The trifurcation-counting mechanism is likely portable to other monotone noisy consensus rules on trees, where 'at least two minus neighbors' is replaced by 'at least $k$ minus neighbors', by adapting the weight $p-2$ in the definition of $\\xi^*$."],"forward_implications":["For every $d \\ge 3$ there is a positive-noise window in which the noisy majority vote process on $\\mathbb{T}_d$ has multiple equilibrium measures, with the all-plus initial configuration retaining a strict plus bias in the limit.","The remaining open regular-tree cases $d = 3$ and $d = 4$ are closed, matching the known $d \\ge 5$ result and the $d = 2$ unique-equilibrium result.","Because the original process dominates the minus biased process, the uniform minus-probability bound transfers directly, so the conclusion concerns the original noisy majority vote dynamics and not only the auxiliary variant.","The same uniform bound holds for any infinite tree of maximal degree $d$, giving a quantitative statement beyond regular trees.","The ergodic/non-ergodic threshold in degree is thereby pinned between $d=2$ and $d=3$: unique equilibrium on the line, multiple equilibria on $\\mathbb{T}_d$ for all $d \\ge 3$."],"supporting_citations":[{"why":"Supplies the previous non-ergodicity result for $d \\ge 5$ on the same process, which this paper extends to $d \\ge 3$.","marker":"[12]"},{"why":"Gives the $d=2$ unique-equilibrium result for the line, the boundary that the new theorem complements.","marker":"[26]"},{"why":"Provides a cellular-automaton counterexample to the positive-rate conjecture, the background motivation for expecting non-ergodicity in noisy monotone dynamics.","marker":"[24]"}],"fun_headline_variants":["Noise loses on tree majority vote for all d ≥ 3","Majority vote noise fails on every tree with degree ≥3","Tree majority vote: non-ergodic for every degree ≥3","On trees, majority vote keeps bias for all d ≥ 3","Every tree degree ≥3: majority vote overrides noise"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire induction rests on Lemma 2.15's tree-specific inequality, which says that for a minimal set $B$ of past minus votes against an odd cluster $A$, the weighted count of branching vertices plus the number of $A$-vertices with two or more neighbors in $B$ is at most $|B|$ minus the number of connected components of $B$.","fun_headline_variants_meta":{"raw":{"variants":["Noise loses on tree majority vote for all d ≥ 3","Majority vote noise fails on every tree with degree ≥3","Tree majority vote: non-ergodic for every degree ≥3","On trees, majority vote keeps bias for all d ≥ 3","Every tree degree ≥3: majority vote overrides noise"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001272,"raw_usage":{"total_tokens":5127,"prompt_tokens":791,"completion_tokens":4336,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":407,"completion_tokens_details":{"reasoning_tokens":4248}},"tokens_in":407,"tokens_out":4336,"duration_ms":27584,"temperature":1.0,"reasoning_tokens":4248,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T13:37:14.531805+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Test Lemma 2.15 directly on a small example: take an odd cluster $A$ in $\\mathbb{T}_3$ and enumerate all minimal even sets $B$ with $\\Prr(A,B) \\neq \\emptyset$; any instance with $\\xi^*(A,B)+|\\Prr(A,B)| > |B|-r$ would refute the lemma. Alternatively, for $d=3$ and a small fixed $\\epsilon$, compute the supremum over $t$ of $\\mathbb{P}(\\sigma_v(t)=-1)$ starting from all plus; a value at or above $1/2$ would falsify Theorem 1.2.","supporting_citations":[{"cited_title":"Bramson and L","cited_arxiv_id":null,"evidence_quote":"Supplies the previous non-ergodicity result for $d \\ge 5$ on the same process, which this paper extends to $d \\ge 3$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the $d=2$ unique-equilibrium result for the line, the boundary that the new theorem complements."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides a cellular-automaton counterexample to the positive-rate conjecture, the background motivation for expecting non-ergodicity in noisy monotone dynamics."}],"review_version":1}