{"id":"bd050f1b-6ae8-4954-ae0f-1a2f5ef421c1","arxiv_id":"2607.10801","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Singular-value gaps of generators and two-point motions control L2 relaxation of non-reversible Markov processes, yielding a proof of the Diaconis–Miclo square-root speedup for lifted walks plus sharp bounds for switching flows and Langevin dynamics.","lead":"The paper gives a systematic way to bound L2 mixing times of non-reversible Markov processes via the singular-value gap of the generator and of its two-point motion. It proves the Diaconis–Miclo square-root speedup conjecture for lifted random walks on abelian groups and supplies sharp bounds for several non-reversible models used in sampling.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader correctly isolates the case-by-case verification of (C2) as the only soft spot, yet that verification is elementary and the resulting constants are of the expected order for the main applications (lifted walks, Langevin, switching flows). The abstract equivalences themselves require no extra assumptions beyond the existence of a core, and the proofs are self-contained. Consequently the ACCEPT verdict with high confidence needs no adjustment.","tokens_in":36240,"tokens_out":362,"duration_ms":6009,"concrete_test":"Independently recompute the constant C_{2}^{2} for the two-point motion of the lifted walk (Lemma 37) by expanding A* A g for a pure Fourier mode g(x,y)=exp(2πi(k·x+\theta·y)/n) and checking that the resulting ratio remains O(d); if the ratio exceeds O(d) the claimed order of t_rel(ε) would need revision, otherwise the bounds stand.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claims rest on clean variational characterizations (Theorems 7, 13, 14) whose proofs are short and elementary, plus an abstract two-component inequality (Theorem 18) whose hypotheses (A1)+(B2)+(C2) are verified by direct computation in every example. The only quantitative gap that appears (extra dimension factors when the two-point motion is used for non-averaged bounds on lifted walks) is already flagged by the authors and does not undermine the equivalence statements or the resolution of the Diaconis–Miclo conjecture. No hidden assumption, circular step, or incorrect estimate was found.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper develops a systematic L^{2} theory for non-reversible Markov processes based on the singular-value gap s(L) of the generator. It proves that 1/s(L) is equivalent (up to universal constants) to the relaxation time of the time-averaged semigroup (Theorem 7), that the singular-value gap of the two-point motion lower-bounds the ordinary spectral gap (Theorem 13) and controls non-averaged relaxation from regular initial laws (Theorem 14), and that an abstract two-component inequality under partial dissipativity (A1), a singular-value gap of the second-order collapse (B2), and a controlled interaction condition (C2) yields explicit upper bounds (Theorem 18 and corollaries). The framework allows non-vanishing first-order collapse and is applied to lifted random walks on abelian groups (resolving the Diaconis–Miclo square-root speed-up conjecture in the form of matching lower bounds on the gap and relaxation time), switching flows with noise, and (perturbed) Langevin dynamics, with sharp upper and lower bounds obtained by direct verification of the abstract conditions.","tokens_in":36400,"tokens_out":1019,"duration_ms":33662,"significance":"The work supplies a clean, broadly applicable replacement for classical spectral-gap and Poincaré methods in the non-reversible setting, together with a simpler alternative to existing hypocoercivity techniques that does not require vanishing first-order collapse. The resolution of the Diaconis–Miclo conjecture for lifted walks on abelian groups is a concrete advance of independent interest; the same machinery recovers (and slightly extends) the best known L^{2} rates for Langevin dynamics with a short proof. All main theorems are proved in full by elementary Hilbert-space arguments and explicit spectral or Bochner estimates; the variational characterizations are parameter-free and the applications reduce to direct verification of (A1)–(C2). These features make the paper a substantial contribution to quantitative Markov-process theory.","major_comments":[{"comment":"The abstract and introduction advertise upper bounds on non-averaged relaxation times via s(L^{(2)}). Theorems 14 and 22 deliver such bounds only after imposing Hilbert–Schmidt regularity on the initial law and after paying logarithmic factors (and, for lifted walks, an extra dimension factor). The gap between the sharp averaged bounds of Theorem 21 and the non-averaged bounds of Theorem 22 is already noted by the authors, but the limitation should be stated more prominently next to the main claims so that readers do not over-interpret the scope of the non-averaged control.","section":null},{"comment":"Theorem 23 confirms the lower bound on gap(L) of order |V|^{-1} sqrt(gap(\\Delta)) and the matching lower bound on t_rel, thereby establishing optimality of the square-root speed-up for relaxation times. The matching upper bound on gap(L) itself (without log factors) remains open with the present method. A short paragraph clarifying precisely which half of Conjecture 1.3 of Diaconis–Miclo is settled, and which half is left open, would strengthen the claim of “proof of a conjecture”.","section":null}],"minor_comments":[{"comment":"Corollary 20(ii): the final display contains “1 + 2C_{1}^{2}/\\gamma^{2}”; the constant should be C_{2} (consistent with the statement of (C2)).","section":null},{"comment":"Equation (1) and the surrounding discussion: the factor e^{2}/(e-1) is elementary but could be recorded once with a one-line proof for completeness.","section":null},{"comment":"Section 3.1, after (16): the explicit constant 16e max(dn^{2} min(1,\\gamma)/20,(1+4d)/\\gamma) is useful; a brief remark that the numerical prefactors are not optimised would prevent readers from treating them as sharp.","section":null},{"comment":"Notation: the same symbol t_rel is used for both averaged and non-averaged times (with and without overline). A consistent typographic distinction throughout (already present in places) would improve readability.","section":null},{"comment":"References: the recent works of Xu (arXiv:2606.01683) and Huang–Li on singular-value gaps could be cross-cited more explicitly when the singular-value gap is introduced.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is a strong fit for a top probability journal. The technical contribution is solid and the Diaconis–Miclo application is a clear selling point. I see no novelty or citation issues."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This one is clean and useful. The main advance is a practical toolkit for L2 relaxation of non-reversible processes: Chatterjee’s singular-value gap controls the time-averaged relaxation time (Theorem 7, elementary), the two-point motion’s singular-value gap lower-bounds the ordinary spectral gap and gives non-averaged bounds from regular initial data (Theorems 13–14), and the first-/second-order collapse framework (Definition 15 + Theorem 18) handles degenerate noise even when the first-order term does not vanish. That last point is what lets them treat continuous-time Markov chains directly and prove the Diaconis–Miclo conjecture for lifted walks on abelian groups (Theorems 21–23), which was open beyond dimension 1.\n\nThe proofs are short Hilbert-space arguments plus direct spectral or Bochner estimates in the examples; Sections 4–5 check out. They recover the sharp Langevin bounds of earlier hypocoercivity work with less machinery (no divergence lemma) and get matching upper/lower orders for the lifted walks and for switching flows. Constants are explicit and the lower bounds come from carefully chosen test functions that capture the position–velocity interaction.\n\nSoft spots are minor and already flagged by the authors. The two-point-motion route for non-averaged relaxation introduces extra dimension factors (visible in the gap between t_rel and t_rel(ε) for the walks), and condition (C2) has to be verified case-by-case; it is not automatic. The framework still only gives averaged or regular-initial-data control in full generality, so pure non-averaged t_rel remains open in some regimes. None of this undercuts the equivalence statements or the conjecture resolution.\n\nThis is for people working on non-reversible MCMC, hypocoercivity, or mixing of lifts. The math is solid, the citations are honest, and there is no circularity. I would send it to referees without hesitation and expect it to be accepted after ordinary polishing.","headline":"Solid, self-contained paper that proves the Diaconis–Miclo square-root speedup and gives a usable collapse framework for non-reversible L2 relaxation.","tokens_in":36963,"tokens_out":513,"would_cite":true,"duration_ms":8148,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60J25","60J27","60J60","47D07"],"pacs":[],"model":"grok-4.5","headline":"Singular-value gaps of generators and two-point motions control L2-relaxation times of non-reversible Markov processes, with a collapse method that proves square-root speed-up for lifted walks.","keywords":["singular value gap","non-reversible","relaxation time","lift","collapse","lifted random walk","two-point motion","hypocoercivity"],"falsifier":"For the lifted random walk on the discrete torus with reorientation rate gamma proportional to 1/n, compute or tightly bound the true L2-relaxation time (averaged or non-averaged) and check whether it is of order n (ballistic) rather than n^2; a substantially larger order would falsify both the collapse bounds and the claimed square-root speed-up.","tokens_in":37177,"feed_emoji":"⏱️","tokens_out":791,"duration_ms":9948,"temperature":0.7,"pith_summary":"For non-reversible Markov processes the usual spectral gap no longer equals the L2-relaxation time, so the paper replaces it by the singular-value gap of the generator. That gap is shown to be equivalent (up to universal constants) to the relaxation time of the time-averaged transition semigroup, and the corresponding gap of the two-point motion lower-bounds the ordinary spectral gap while also controlling non-averaged relaxation from regular initial laws. To obtain concrete lower bounds on these gaps when the noise is degenerate, the authors introduce first- and second-order collapses of the generator onto a low-frequency subspace; under three abstract conditions (partial dissipativity, a gap for the second-order collapse, and a controlled interaction) they derive explicit upper bounds on the averaged relaxation time. The method is simple enough to cover continuous-time Markov chains as well as diffusions and piecewise-deterministic processes, and it yields sharp constants for lifted random walks on abelian groups (confirming a square-root speed-up conjecture), for switching flows, and for Langevin dynamics.","feed_headline":"Singular-value gaps control non-reversible mixing times","feed_subtitle":"A collapse method proves square-root speed-up for lifted walks and bounds Langevin dynamics","key_machinery":"The singular-value gap s(L) = inf ||Lf||/||f|| together with the first- and second-order collapses C1, C2 of L onto a low-frequency subspace; the three abstract conditions (A1) partial dissipativity, (B2) gap of C2, and (C2) controlled interaction via the remainder operator A turn these collapses into explicit bounds via a lifted Poincaré inequality.","core_discovery":"The inverse of the singular-value gap s(L) is equivalent, up to universal constants, to the L2-relaxation time of the time-averaged semigroup. Moreover the singular-value gap of the two-point motion L^(2) lower-bounds the ordinary spectral gap of L and controls non-averaged relaxation from regular initial measures. Under the three conditions (A1)+(B2)+(C2) on first- and second-order collapses one obtains matching upper bounds on these quantities.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Singular-value gaps set L2-relaxation for non-reversible Markov processes","Collapse method proves square-root speed-up for lifted random walks","Two-point singular gaps bound ordinary spectral gaps of generators","First- and second-order collapses yield sharp non-reversible mixing bounds","Singular gaps control non-averaged relaxation from regular initial laws"],"cache_read_input_tokens":32896,"weakest_assumption_plain":"The three abstract conditions on partial dissipativity, the second-order collapse gap, and especially the controlled interaction between high- and low-frequency subspaces must hold with constants of the right order; they are verified case-by-case and can introduce extra dimension factors that are not always sharp.","fun_headline_variants_meta":{"raw":{"variants":["Singular-value gaps set L2-relaxation for non-reversible Markov processes","Collapse method proves square-root speed-up for lifted random walks","Two-point singular gaps bound ordinary spectral gaps of generators","First- and second-order collapses yield sharp non-reversible mixing bounds","Singular gaps control non-averaged relaxation from regular initial laws"]},"model":"grok-4.5","effort":"low","cost_usd":0.005968,"raw_usage":{"total_tokens":1604,"prompt_tokens":818,"num_sources_used":0,"completion_tokens":94,"cost_in_usd_ticks":59680000,"prompt_tokens_details":{"text_tokens":818,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":692,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":818,"tokens_out":94,"duration_ms":8083,"temperature":1.0,"reasoning_tokens":692,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-14T09:08:44.399417+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"For the lifted random walk on the discrete torus with reorientation rate gamma proportional to 1/n, compute or tightly bound the true L2-relaxation time (averaged or non-averaged) and check whether it is of order n (ballistic) rather than n^2; a substantially larger order would falsify both the collapse bounds and the claimed square-root speed-up.","supporting_citations":[],"review_version":1}