{"id":"d943edb1-1067-4831-8253-316e2552bacb","arxiv_id":"2411.13964","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A rigorous proof that lattice run-and-tumble particle models converge to a continuous jamming piecewise deterministic Markov process, with sharp non-asymptotic mixing time bounds.","lead":"Two run-and-tumble particles on a lattice are shown to converge, as the lattice spacing shrinks, to a continuous model on a circle where the particles jam when they meet. The paper also determines, up to constants, how long the system takes to reach its stationary behavior, revealing fast and slow regimes.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Missing lower-bound proof for CITP mixing time: Section 4.1 only proves CFTP lower bounds, leaving Theorem 3(i) half-unverified.","rationale":"The paper's core convergence claims are well supported: the pL mapping in Definition 5 is carefully justified for DITP/DFTP, the martingale estimates in Lemmas 1 and 2 are coherent, and Proposition 5's uniformity argument is sound modulo the stated 'slight modifications' for DFTP. The upper mixing-time bounds are derived from explicit hitting-time computations (Lemma 8) and return-time concentration (Lemma 11), which are internally consistent. The weakest point is the unproved lower bound for the CITP in Theorem 3(i), explicitly acknowledged by the text's decision to prove only assertion (ii). Because the theorem advertises matching upper and lower bounds, a missing lower bound leaves the central scaling claim half-unsupported. The reader's weakest_assumption focused on generality of tumble mechanisms, but that is not a flaw in the stated theorem, which is explicitly restricted to DITP/DFTP; the more load-bearing issue is the omitted CITP lower-bound argument. The conditional verdict remains appropriate: no demonstrated error, but the proof is incomplete as written.","tokens_in":34097,"tokens_out":42143,"duration_ms":410250,"concrete_test":"Write out the CITP lower-bound proof: adapt Lemma 6 to the two-state velocity process by computing the MGF of the integrated velocity via the 2x2 generator and proving a Kolmogorov-type maximal inequality; then verify from Proposition 7 that for M=((1-a)ell/2,(1+a)ell/2)xSigma, pi(M)<=a, using the explicit bulk density omega/(2+omega ell) and boundary atoms in Table 1. If both steps yield C(epsilon)(1/omega)(1+omega^2 ell^2) <= t^i_mix(epsilon), Theorem 3(i) is complete; if the central-interval estimate fails for some omega ell, the diffusive lower bound must be revised.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 4.1 states 'only the lower bound of assertion (ii) is proven' and then concludes assertion (ii) from Lemmas 5 and 7; the lower bound in Theorem 3(i) for the CITP is never written. The claimed optimal scaling t^i_mix(epsilon)=Theta((1/omega)(1+omega^2 ell^2)) therefore depends on an unstated 'identical arguments' claim. This is not a routine transcription: Lemma 6 relies on the three-state velocity structure (excursions to 0, i.i.d. D_i, Kolmogorov inequality) and Lemma 7 uses the CFTP invariant measure; neither has a stated CITP analogue. In particular, the diffusive lower bound needs (a) a two-state version of Lemma 6 with t*=u(epsilon) omega delta^2, and (b) the estimate pi(M^c)>=1-a for the CITP invariant measure of Proposition 7 (central interval mass <=a). Since the lower bound is half of the optimality statement, its omission is load-bearing. This is a completeness gap, not a demonstrated error.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper provides a rigorous connection between two discrete-space run-and-tumble particle models with jamming (DITP and DFTP) and their continuous-space PDMP limits (CITP and CFTP). Under the scaling gamma_L=(L-1)/ell, the authors construct an explicit coupling and prove a quantitative Skorokhod-convergence bound (Theorem 2), use a uniform exponential ergodicity result for the discrete models (Proposition 5) to deduce Wasserstein convergence of invariant measures (Corollary 1), derive exact stationary measures with Dirac masses at jammed configurations and exponential bulk densities (Propositions 7-8), and give matching upper and lower bounds for the mixing times of the CITP and CFTP (Theorem 3), of orders (1/omega)(1+omega^2 ell^2) and (1/alpha+1/beta)(1+alpha^2 ell^2). The proofs are largely self-contained and use couplings, Doeblin minorization, deterministic hitting lemmas, first-step analysis, and explicit ODE solutions.","tokens_in":34298,"tokens_out":15400,"duration_ms":143043,"significance":"If the gaps identified below are repaired, this is a useful contribution. It justifies the previously formal continuous limits in [SEB16, SEB17], provides explicit invariant measures for a non-reversible interacting active-particle system, and gives non-asymptotic mixing-time estimates with the correct dependence on the model parameters. The paper's self-contained coupling method, including the lumping construction of Definition 5 and the hitting-time ODE calculations, is a strength; it avoids spectral assumptions and yields explicit probability bounds. The explicit scaling predictions for the persistent and diffusive regimes are falsifiable and should be of interest to the active-matter community. The candid discussion of the limitations of the coupling approach for general tumble mechanisms (Remark 2) is also a positive feature.","major_comments":[{"comment":"Theorem 3(i) is not proven as stated. Section 4.1 explicitly says that \"only the lower bound of assertion (ii) is proven\" and then proves Lemmas 5-7 for the CFTP. The lower bound for the CITP, t^i_mix(epsilon) >= C(epsilon)(1/omega)(1+omega^2 ell^2), requires a two-state analogue of Lemma 6, whose excursion decomposition uses the zero-velocity state and does not transfer verbatim to the +/-1-only CITP velocity chain, and also requires an invariant-measure estimate pi(M^c) >= 1-a for the CITP analogous to the one used in Lemma 7. Since Theorem 3(i) is the paper's optimality claim for the CITP, this is a load-bearing gap rather than a presentational shortcut. The gap appears repairable by a parallel argument, but the proof needs to be written.","section":"Section 4.1, Theorem 3(i)"},{"comment":"Lemma 12 is stated without proof (\"the proof ... omitted for the sake of brevity\") and is then used in Proposition 6 to establish the symmetries pi = rho_i#pi of the invariant measure. Those symmetries are used in Appendix A.2 to fix the constants in the explicit invariant measures (c2 = e^{kappa ell} c1, c4 = 0, and the relations d^ell_{(sigma1,sigma2)} = d^0_{(sigma2,sigma1)}), which in turn support Theorem 1(3) and the lower bound Lemma 7. As written, the appendix's derivation of the invariant measure is therefore incomplete. Please provide the proof of Lemma 12 or give a precise reference to the corresponding statement in [HGM25] and explain how it applies to the present construction.","section":"Appendix A.1, Lemma 12"},{"comment":"The uniformity result for the DFTP is not proved in the manuscript. Lemma 4(i)-(ii) is proved only for the DITP; the text says the same arguments apply \"up to some slight modifications\" to the DFTP. Proposition 5, which supplies the t -> infinity and L -> infinity interchange for both processes and hence Corollary 1 for the DFTP, depends on Lemma 4(ii) for the DFTP. Please include the DFTP calculation (the generating function for the three-state velocity chain and the resulting fourth-moment ratio) or provide a reference where this is done.","section":"Section 3.2, Lemma 4 / Proposition 5"}],"minor_comments":[{"comment":"The conclusion of Lemma 7 reads \"ti_mix(epsilon) >= ...\" but the proof is for the CFTP; it should read t^f_mix(epsilon).","section":"Section 4.1, Lemma 7"},{"comment":"The minorization state is written as xi = epsilon delta_{(0,(-1,1))}, while the statement and proof use (0,(1,-1)); the two are distinct states, so the Dirac should be at (0,(1,-1)).","section":"Proposition 4"},{"comment":"The notation for the factor (-1)^i sigma_i is typeset without the exponent in several displays (e.g., in Definition 5 and Lemma 1), which makes expressions such as \"(-1)i sigma_i\" hard to parse; please fix the typesetting.","section":"Throughout"},{"comment":"There are several typos, including \"proccess\" (Proposition 4), \"identfies\" (Lemma 2), \"unnormalizedinvariant\" (Proposition 8), and \"a pairs\" (Section 1.2.1).","section":"Various locations"}],"recommendation":"major_revision","confidential_remarks":"The paper is technically serious and the coupling apparatus is credible. I regard the missing CITP lower bound as the decisive issue; it is a proof gap rather than a demonstrated error, and I expect it can be supplied by a parallel argument. The omitted proof of Lemma 12 should also be supplied. I do not see grounds for rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis is the proof that the on-lattice RTP models of Slowman-Evans-Blythe converge to the continuous PDMP, with quantitative coupling bounds. That was widely assumed and now it is actually shown. The mixing time results are also new and the persistent/diffusive regime picture is convincing. The paper is worth taking seriously.\n\nThe core convergence argument (Theorem 2) is solid: the coupling via the periodic map p_L in Definition 5 is a genuine trick, and the bound with the n(T) factor is explicit. Proposition 5, the uniformity that gives Wasserstein convergence of invariant measures, is properly proved via the discrete-discrete coupling and the variation lemmas. The upper bounds on mixing times are built on explicit hitting time ODEs, and the CFTP lower bound is a substantial argument.\n\nNow the soft spots. Lemma 12 is stated without proof, and it is used to get the symmetries of the invariant measure, which in turn feed the explicit formulas used in the lower bounds. The authors say the proof is omitted for brevity; it is probably a simple commutation, but in a paper that leans on this chain, it should be written out or moved to an appendix. Bigger issue: the lower bound in Theorem 3(i) for the CITP is never proven. Section 4.1 explicitly says only the lower bound of assertion (ii) is proven, then asserts (i) by 'identical arguments.' The stress-test note is right that this is not a routine transcription: the CFTP lower bound uses the three-state velocity structure and the CFTP invariant measure, and the CITP needs its own two-state version plus the CITP invariant mass estimate. Without that, the optimality claim for the CITP mixing time is half-verified. This is a completeness gap, not a demonstrated error.\n\nAlso Proposition 7's proof is skipped as 'similar,' which is fine if the referee gets the details, but combined with Lemma 12 there is a pattern of deferred verification.\n\nOverall: the central theorems are supported by real arguments, the gaps are addressable, and the paper is a genuine advancement. Send it to a serious referee with instructions to demand the missing proofs rather than accept them on faith.","headline":"Genuine discrete-to-continuous convergence proof for RTP jamming models, but the CITP lower bound and two lemmas are left unproved.","tokens_in":34840,"tokens_out":2184,"would_cite":true,"duration_ms":24409,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60J25","60J27","60B10","82C22"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that two run-and-tumble particles with jamming have a well-defined continuous limit, and pins down the time they take to reach their stationary regime.","keywords":["run-and-tumble particles","jamming","piecewise deterministic Markov processes","mixing time","invariant measure","scaling limit","active matter","motility-induced phase separation"],"falsifier":"Simulate the DITP and DFTP on a ring of $L$ sites with $\\gamma_L=(L-1)/\\ell$ together with the coupled continuous processes CITP and CFTP, and check the coupling bound of Theorem 2: if the supremum distance $\\sup_{t\\le T}|i_L(y_L(t))-x(t)|$ fails to tend to zero at the stated rate, or if the empirical invariant measure does not approach the continuous measure in Wasserstein distance as $L\\to\\infty$, the discrete-to-continuous claim would be false. For the mixing-time claim, measure the total-variation distance to stationarity for large $\\omega\\ell$: if the relaxation time does not grow like $\\omega\\ell^2$ (respectively $\\alpha^2\\ell^2$) up to constants, the matching bounds would be contradicted.","tokens_in":33913,"feed_emoji":"🏃","tokens_out":7299,"duration_ms":60727,"temperature":0.7,"pith_summary":"This paper establishes a rigorous bridge between two ways of describing a pair of run-and-tumble particles (RTPs) that cannot pass through each other: a discrete lattice model with blocking and a continuous-space model on a ring. The discrete instantaneous and finite tumble processes DITP and DFTP are shown to converge, under the scaling $\\gamma_L=(L-1)/\\ell$ and with explicit coupling bounds, to continuous piecewise deterministic Markov processes CITP and CFTP. The same convergence transfers to their invariant measures in Wasserstein distance, and the continuous invariant measure has positive mass at the jammed configurations plus exponential decay away from them, which the authors read as effective attraction consistent with motility-induced phase separation. In addition, the paper proves matching upper and lower bounds for the mixing times, of order $(1/\\omega)(1+\\omega^2\\ell^2)$ for the instantaneous-tumble process and $(1/\\alpha+1/\\beta)(1+\\alpha^2\\ell^2)$ for the finite-tumble process, exposing persistent and diffusive regimes.","feed_headline":"Jammed run-and-tumble pairs: lattice converges to continuous","feed_subtitle":"Discrete blocking RTPs match a continuous process; mixing times are optimal in the parameters.","key_machinery":"The argument is carried by the PDMP construction of Davis, in which the continuous process is described on a state space that adds explicit boundary states representing jammed contacts, so the generator has transport terms in the bulk and pure jump terms at jamming. For the discrete-to-continuous limit, the central device is a coupling that maps the jamming dynamics back to a jamming-free random walk: when both velocities are equal ($\\sigma_1=\\sigma_2=\\pm 1$), the separation process is represented through the periodic map $p_L$, whose lumping property lets the authors transfer random-walk fluctuation estimates to the blocked process. Long-time behavior is controlled by an order-preserving coupling between two copies of the process, reducing the mixing time to hitting times of the jammed states; those hitting times are computed from the generator through systems of differential equations, while matching lower bounds come from identifying slow observables and applying concentration inequalities such as Paley-Zygmund.","core_discovery":"The central claim is that the on-lattice jump processes DITP and DFTP, obtained by putting two RTPs on a periodic chain with jamming exclusion, have a well-defined continuous limit: under $\\gamma_L=(L-1)/\\ell$, the rescaled processes converge in the Skorokhod topology to the CITP and CFTP, which are PDMPs on the torus whose generator is constructed through Davis's formalism with explicit boundary states for jamming. The paper proves this convergence by an explicit coupling rather than by generator methods, which gives quantitative control and also yields Wasserstein convergence of the discrete invariant measures to the continuous one. It further establishes that the continuous invariant measures put positive mass at the jammed separations $0$ and $\\ell$, with density decaying exponentially in the bulk, and that the mixing times of the continuous processes are, up to constants depending only on the target accuracy, $(1/\\omega)(1+\\omega^2\\ell^2)$ and $(1/\\alpha+1/\\beta)(1+\\alpha^2\\ell^2)$; the lower bounds match the upper bounds, so the dependence on $\\omega, \\alpha, \\beta, \\ell$ is optimal.","pith_inferences":["A generator-based proof would likely extend the scaling limit to the general tumble mechanisms treated in the companion PDMP framework, but at the cost of quantitative rates; the boundary-jamming states make the generator domain delicate, so this is a natural next step rather than an immediate consequence.","The $p_L$ mapping suggests a lumpability criterion: any tumble mechanism whose conditioned separation process is lumpable to the interval process should admit the same style of quantitative convergence, a property that could be checked by verifying the associated transition-rate identity.","The mixing-time formula predicts that in the diffusive regime the position is the slowest observable and behaves diffusively, so the $\\omega\\ell^2$ term should be recoverable from a spectral-gap calculation for large $\\ell$; a direct check on the generator's eigenvalues would sharpen the constant.","The singular nature of the jamming mass means the effective attraction is not a smooth potential; extending the result to $N>2$ particles will require handling multiple simultaneous jammed contacts, which the pair analysis leaves open."],"forward_implications":["The lattice-to-continuum approximation used in earlier exact solutions of jamming RTPs is valid: as the lattice spacing vanishes, the discrete processes converge in Skorokhod topology and their invariant measures converge in Wasserstein distance.","The continuous invariant measure has positive mass at jammed separations $0$ and $\\ell$ and decays exponentially away from them, so a pair of RTPs exhibits effective attraction without any interaction energy, matching the microscopic picture of motility-induced phase separation.","The CITP mixing time is of order $(1/\\omega)(1+\\omega^2\\ell^2)$: linear in $1/\\omega$ in the persistent regime $\\omega\\ell\\ll 1$, and diffusive $\\omega\\ell^2$ in the regime $\\omega\\ell\\gg 1$.","The CFTP mixing time is of order $(1/\\alpha+1/\\beta)(1+\\alpha^2\\ell^2)$, and it reduces to the CITP result when $\\beta\\to\\infty$ with $\\omega=\\alpha/2$.","The upper and lower bounds on mixing times match, so the parameter dependence is optimal and the accuracy-dependent constant behaves like $\\log(1/\\epsilon)$ for small $\\epsilon$.","The quantitative coupling bounds give non-asymptotic control on the approach to stationarity, going beyond spectral asymptotics for these non-reversible processes."],"supporting_citations":[{"why":"Defines the discrete instantaneous tumble process DITP and computes its invariant measure; this is the process whose continuous limit is established.","marker":"[SEB16]"},{"why":"Defines the finite tumble process DFTP and its invariant measure, which the paper shows converges to the CFTP invariant measure.","marker":"[SEB17]"},{"why":"Provides the general PDMP framework for two jammed RTPs, the explicit invariant measures and the universality classes that the continuous limit recovers.","marker":"[HGM25]"},{"why":"Supplies the PDMP construction and generator characterization used to define CITP and CFTP and to compute hitting times.","marker":"[Dav93]"},{"why":"Provides the lumpability theorem used to prove that the $p_L$-mapped process has the DITP/DFTP transition rates.","marker":"[BY93]"},{"why":"Motivates the coupling approach yielding non-asymptotic mixing-time bounds for non-reversible processes.","marker":"[FGM12]"},{"why":"Supplies the order-preservation argument reducing the coupling problem to hitting time computations.","marker":"[LMT96]"},{"why":"Gives the Doeblin and minorization criteria used to prove uniqueness, exponential ergodicity, and uniformity over the lattice size.","marker":"[BH22]"},{"why":"Provides the Skorokhod-space criterion used to turn the quantitative sup-norm coupling bounds into convergence in law.","marker":"[Bil99]"}],"fun_headline_variants":["Run-and-tumble jamming: lattice to continuous limit proven","Discrete RTPs converge to jammed continuous process","Optimal mixing times for run-and-tumble pairs","Jammed RTPs: discrete and continuous match exactly","Lattice run-and-tumble converges to continuous model"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof leans on the special structure of the two velocity mechanisms: when $\\sigma_1=\\sigma_2=\\pm 1$, the separation process can be written through the periodic map $p_L$, and without such a map the quantitative coupling bound does not go through, so the convergence theorem is established only for the instantaneous-tumble and finite-tumble mechanisms of figures 2a and 2b, not for general tumble rates.","fun_headline_variants_meta":{"raw":{"variants":["Run-and-tumble jamming: lattice to continuous limit proven","Discrete RTPs converge to jammed continuous process","Optimal mixing times for run-and-tumble pairs","Jammed RTPs: discrete and continuous match exactly","Lattice run-and-tumble converges to continuous model"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000252,"raw_usage":{"total_tokens":1616,"prompt_tokens":1058,"completion_tokens":558,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":674,"completion_tokens_details":{"reasoning_tokens":478}},"tokens_in":674,"tokens_out":558,"duration_ms":64595,"temperature":1.0,"reasoning_tokens":478,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T15:42:06.847690+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate the DITP and DFTP on a ring of $L$ sites with $\\gamma_L=(L-1)/\\ell$ together with the coupled continuous processes CITP and CFTP, and check the coupling bound of Theorem 2: if the supremum distance $\\sup_{t\\le T}|i_L(y_L(t))-x(t)|$ fails to tend to zero at the stated rate, or if the empirical invariant measure does not approach the continuous measure in Wasserstein distance as $L\\to\\infty$, the discrete-to-continuous claim would be false. For the mixing-time claim, measure the total-variation distance to stationarity for large $\\omega\\ell$: if the relaxation time does not grow like $\\omega\\ell^2$ (respectively $\\alpha^2\\ell^2$) up to constants, the matching bounds would be contradicted.","supporting_citations":[],"review_version":1}