{"id":"113b3115-df37-4535-b459-46408b86a620","arxiv_id":"2606.05386","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Vertex-reinforced walks with geometric holding times converge almost surely to the same fixed points of x=π(x) as the simultaneous-transition model, via a martingale-plus-geometric-decay decomposition that restores the Clark-Kushner condition.","lead":"Interacting reinforced random walks that jump at independent geometric times still have the same almost-sure occupation limits as the classical simultaneous-jump model. The result shows that asynchronous jump rates do not change the equilibria of the occupation measure, only the path to them.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The Reader correctly isolates the external Lyapunov hypothesis as the weakest modelling assumption while recognising that the paper’s genuine technical contribution—the verification of (8) under geometric holding times—is complete and free of circularity. My own reading of the ten-step proof finds no internal inconsistency or missing estimate that would undermine Theorem 2. Consequently the ACCEPT verdict stands; the only residual risk is the usual one that Assumption 2 may fail for some concrete π of interest, which is already flagged by the Reader and does not affect the correctness of the argument as written.","tokens_in":11899,"tokens_out":494,"duration_ms":6701,"concrete_test":"Independently re-derive the bound on term (I) in Step 9: expand c_j^r=γ_j/p-d_j^r, verify that both pieces of d_j^r are O(1/j^{2}) or geometrically small, and confirm that the resulting series still vanish uniformly in r as n\to∞. If the uniform Cauchy property of the weighted martingale fails for any p∈(0,1], the Clark–Kushner verification collapses.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (a.s. convergence of X(n) to F^{-1}(0) under Assumptions 1–2) rests on verifying the Clark–Kushner condition (8) for the biased noise U(n). The paper supplies a complete, self-contained decomposition U(n)=D(n)+(1-p)U(n-1)+E(n-1) (eq. 15) and controls each of the three resulting sums (I)–(III) by martingale convergence plus geometric decay. The estimates (22)–(27) appear tight and use only the Lipschitz property of π, the bound ∥U∥≤2, and ∑γ_n^{2}<∞; no hidden gap is visible. The Lyapunov assumption (Assumption 2) is external and standard for the Benaïm framework, exactly as the Reader notes, but it is not required for the novel part of the argument (Theorem 2).","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper studies a system of m interacting vertex-reinforced random walks on a finite complete graph with d vertices, in which each walk i jumps at independent geometric inter-transition times of parameter p_i in (0,1]. The occupation-proportion vector X(n) evolves as a stochastic approximation X(n+1)-X(n)=\\gamma_n(F(X(n))+U(n+1)) with F(x)=-x+\\pi(x). Because the one-step matrices are of the form Q^i(x,p_i)=p_i \\Pi^i(x)+(1-p_i)I, the invariant measures remain \\pi^i(x) independently of the p_i, so the candidate limit set is still F^{-1}(0). The main technical contribution is a verification of the Clark–Kushner condition (8) for the biased noise U(n)=\\xi(n)-\\pi(X(n-1)): the authors decompose U(n)=D(n)+(1-p)U(n-1)+E(n-1) into a martingale difference, a geometrically decaying term, and a Lipschitz-controlled remainder, then show that each of the three resulting weighted sums vanishes uniformly on compact time windows. Under the standing Lipschitz and strict-Lyapunov assumptions, the accumulation points of X(n) therefore lie in F^{-1}(0) (and X(n) converges a.s. when that set is countable).","tokens_in":12112,"tokens_out":951,"duration_ms":28614,"significance":"The result shows that the almost-sure limit set of the joint occupation measure is robust to asynchronous geometric holding times; the same equilibria appear as in the classical simultaneous-jump models of Rosales–Prado–Pires and Prado–Rosales. The proof technique—explicit solution of the linear recurrence induced by the geometric bias and control of the three series by martingale convergence plus geometric decay—is self-contained, uses only standard tools (Lipschitz continuity of \\pi, square-summability of \\gamma_n, and the algebraic structure Q=p\\Pi+(1-p)I), and is potentially reusable for other stochastic-approximation schemes whose noise is a convex combination of the target and the current state. The paper therefore supplies a clean, non-trivial extension of the existing interacting-VRRW literature.","major_comments":[],"minor_comments":[{"comment":"Abstract and Introduction: the phrase “establishing almost sure convergence” slightly overstates Theorem 1, which only guarantees that the connected set of accumulation points lies inside F^{-1}(0) and that a.s. convergence to a single point holds when that set is countable. Align the wording with the precise statement of Theorem 1.","section":null},{"comment":"Section 3.1 (reduction to m=1): while the argument is component-wise and extends immediately to distinct p_i, a one-sentence remark that the same estimates hold with max_i(1-p_i) (or component-wise) would make the multi-walk case fully explicit.","section":null},{"comment":"Step 8, display after (23): the inequality \\gamma_{j-1}\\gamma_{j+1}\\le2\\gamma_j^{2} is correct for the chosen ℓ^{1}-norm and γ_n=1/(n+2), but a brief verification (or a uniform constant C) would improve readability for small j.","section":null},{"comment":"Typographical: author name “Grac ¸ adio”, repeated spacing/encoding artefacts around accents (Benaïm, etc.), and the arXiv date “June 5, 2026” should be cleaned before publication.","section":null},{"comment":"Assumption 2 is imported from Benaïm’s framework without a concrete example; a short pointer to a family of maps π for which a strict Lyapunov function is known (e.g., from the cited works) would help the reader.","section":null}],"recommendation":"accept","confidential_remarks":"Solid, self-contained contribution that fits well in a probability journal. The technical core (verification of Clark–Kushner for geometrically biased noise) is cleanly executed and of independent interest. No concerns about novelty disclosure or citation pattern."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The punchline is simple: once you let each walk jump on its own geometric clock, the occupation measure still converges a.s. to the same zeros of F(x) = -x + π(x) that appear in the simultaneous-jump models. The technical work is the decomposition that kills the bias so Clark–Kushner still holds.\n\nWhat is new is exactly that decomposition. Because Q = pΠ + (1-p)I, the one-step expectation is a convex combination of π(X(n)) and the current state. The authors write U(n) = D(n) + (1-p)U(n-1) + E(n-1), solve the linear recurrence, and show that the three resulting series vanish uniformly on compact time windows. The martingale piece converges by the usual square-summability argument; the geometric factor (1-p) turns the other two pieces into O(γ_n) or O(∑ γ^{2}) terms. The estimates are elementary, use only Lipschitz of π and the crude bound ∥U∥ ≤ 2, and look tight. No hidden gap is visible on a careful read.\n\nThe paper does the bookkeeping carefully and writes the argument out in full (Steps 1–10). That is useful; the simultaneous-jump papers do not supply this calculation. The fixed-point set is independent of the p_i’s, which is the modelling takeaway.\n\nSoft spots are minor and standard for the literature. Assumption 2 (existence of a strict Lyapunov function for the ODE) is imported from Benaïm and left unverified for concrete π; if it fails, the accumulation-point claim collapses even when Clark–Kushner holds. That is not a flaw in the novel part (Theorem 2). The result is incremental: it removes a modelling restriction that appears in several recent papers by the same group, but does not introduce a new general method or settle an open conjecture. No free parameters, no circularity, citations look appropriate.\n\nThis is for people already working on interacting reinforced processes or stochastic approximation with state-dependent noise. A serious referee should see it; the math is solid enough to deserve that time. I would accept for peer review and would cite the decomposition if I ever need asynchronous clocks in this setting.","headline":"Clean, self-contained extension of interacting VRRW to geometric clocks; the bias decomposition works and the limiting set is unchanged.","tokens_in":12703,"tokens_out":611,"would_cite":true,"duration_ms":5957,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","60F15","37C10"],"pacs":[],"model":"grok-4.5","headline":"Geometric holding times do not change the almost-sure limits of interacting vertex-reinforced walks.","keywords":["reinforced random walk","stochastic approximation","geometric inter-transition times","vertex occupation measure","Clark-Kushner condition","interacting walks"],"falsifier":"Construct an explicit smooth reinforcement map π on a small complete graph for which the associated vector field F has no strict Lyapunov function (or whose only Lyapunov function is infinite on some equilibria) and check whether the occupation process still converges; failure of convergence would falsify the claimed application of the theorem.","tokens_in":12818,"feed_emoji":"🎲","tokens_out":678,"duration_ms":5204,"temperature":0.7,"pith_summary":"The paper studies several interacting random walks on a finite graph whose next-vertex probabilities depend on the whole history of visits made by all of them. Unlike earlier models in which every walk moves at every discrete time, each walk here waits a geometric number of steps with its own success probability before it may jump. The authors show that the vector of long-run occupation proportions still converges almost surely to the same set of equilibria that appears when every walk is forced to move simultaneously. Those equilibria are simply the fixed points of the map that sends current occupation proportions to the preferred next-vertex distributions. The technical obstacle is that the usual martingale argument of stochastic approximation fails because a walk that does not jump stays put, introducing a state-dependent bias; the paper removes the obstacle by writing the bias as a martingale plus a geometrically decaying remainder.","feed_headline":"Geometric waits leave reinforced-walk limits unchanged","feed_subtitle":"Occupation measures still converge to the same fixed points as when every walker moves every step","key_machinery":"The recursive decomposition U(n) = D(n) + (1−p)U(n−1) + E(n−1) of the stochastic-approximation noise, where D is a martingale difference and the factor (1−p)<1 supplies geometric decay that converts the otherwise divergent weighted sums into convergent series, thereby verifying the Clark–Kushner condition.","core_discovery":"Under standard Lipschitz and Lyapunov assumptions on the reinforcement map, the occupation-measure process of the interacting walks converges almost surely to the zero set of the vector field F(x) = −x + π(x). Because the one-step transition matrix of each walk has the form p_i Π^i(x) + (1−p_i)I, its unique invariant measure is exactly π^i(x) and is therefore independent of the geometric parameter p_i. Consequently the possible limit points remain exactly the same as in the classical simultaneous-transition model.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Geometric waits keep reinforced-walk limits the same","Occupation limits match simultaneous model under geometric waits","Vertex-reinforced walks share fixed points with geometric delays","Geometric inter-transition times leave limit set unchanged","Reinforced walks converge to same points despite geometric waits"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The continuous-time flow generated by the mean-field vector field must admit a strict Lyapunov function that is finite on the set of equilibria; without that Lyapunov function the stochastic-approximation theorem does not guarantee that accumulation points lie inside the zero set.","fun_headline_variants_meta":{"raw":{"variants":["Geometric waits keep reinforced-walk limits the same","Occupation limits match simultaneous model under geometric waits","Vertex-reinforced walks share fixed points with geometric delays","Geometric inter-transition times leave limit set unchanged","Reinforced walks converge to same points despite geometric waits"]},"model":"grok-4.5","effort":"low","cost_usd":0.003324,"raw_usage":{"total_tokens":1158,"prompt_tokens":810,"num_sources_used":0,"completion_tokens":55,"cost_in_usd_ticks":33240000,"prompt_tokens_details":{"text_tokens":810,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":293,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":810,"tokens_out":55,"duration_ms":5020,"temperature":1.0,"reasoning_tokens":293,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-14T18:25:19.083470+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Construct an explicit smooth reinforcement map π on a small complete graph for which the associated vector field F has no strict Lyapunov function (or whose only Lyapunov function is infinite on some equilibria) and check whether the occupation process still converges; failure of convergence would falsify the claimed application of the theorem.","supporting_citations":[],"review_version":2}