{"id":"3c23a3e1-c39a-4f59-b0cb-df14b1636d4a","arxiv_id":"2506.12632","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For K-symmetric exclusion from a step profile, the rescaled point process of extreme particles converges to a Poisson random measure with intensity proportional to e^{-x} dx.","lead":"This paper proves that the rightmost particles in a many-particle exclusion model become randomly scattered like a Poisson point process at large times. The result generalizes a known Gumbel limit for simple exclusion to any occupancy limit K and gives joint laws for leading particle positions and gaps.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proof is coherent; no internal flaw found. The sole load-bearing external dependency is Lemma 5.1 from [11]; its hypotheses for indicator functions should be verified.","rationale":"The reader's strongest claim, Theorem 3.4, is supported by a coherent proof structure. The factorial-moment method is applied correctly via Lemmas 2.1-2.4, the mean convergence in Proposition 4.1 follows from the Cesàro condition and the sharp estimates in [6], and the error bounds in Propositions 4.3-4.5 all trace back to the semigroup comparison Lemma 5.1. I found no internal mathematical inconsistency: the indicator functions used are positive definite in the stated sense, the set inclusions in Corollary 5.7 are valid, and the error terms decay at the required rates. The one genuinely load-bearing point is the external Lemma 5.1; the paper neither proves it nor checks its hypotheses in detail for the indicator functions of unbounded sets used later. This is a standard reliance on a published result rather than a demonstrated flaw, so I would not change the accept verdict. I partially agree with the reader's identification of Lemma 5.1 as the weakest assumption: it is the unique external input that could invalidate the correlation control if misstated, but no evidence of misstatement was found. The proposed test would settle whether the dependency is correctly imported.","tokens_in":31725,"tokens_out":33392,"duration_ms":349807,"concrete_test":"Check the exact theorem in [11] (Correlation Inequalities for Interacting Particle Systems with Duality, 2010) that underlies Lemma 5.1, and verify that its hypotheses are satisfied by f = 1_{A^n} for A = v_t^{-1}(a,b] and A = (-∞,0], with the n-particle K-SEP semigroup. In particular, confirm that the positive definite class in [11] includes indicator functions of finite intervals and half-lines, and that the inequality holds pointwise on Ω_K^n for all t ≥ 0. If the original theorem requires f to have finite support or excludes unbounded sets, then the proof of Lemma 5.6 and Proposition 4.4(b) would need a new argument. As a complementary check, perform exact diagonalization of V_K^2(t)1_{A×A} and U_K^2(t)1_{A×A} for K=2, A=[-N,N] with N=1,2,3, at several times t, to numerically confirm the inequality for small systems.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The factorial-moment convergence in Theorem 3.4 hinges on the semigroup comparison Lemma 5.1, imported from [11], which states V_K^n(t)f ≤ U_K^n(t)f for symmetric positive definite f on Ω_K^n. This lemma is used in Corollary 5.2, Lemma 5.6, Corollary 5.7, and ultimately in Propositions 4.3 and 4.5 to control the correlation errors κ_t and τ_t. The applications require f to be an indicator 1_{A^n} where A is a finite interval or a half-line, e.g., A = v_t^{-1}(a,b] or A = (-∞,0]. However, the paper defines positive definiteness only for functions on Z^n, while Lemma 5.1 is stated for functions on the subset Ω_K^n, and no proof or detailed hypothesis check is provided. If [11]'s theorem requires f to be a cylinder function, to have finite support, or if the inequality does not extend to indicators of unbounded sets, then the upper bounds in Corollary 5.7 and hence the error estimates in Proposition 4.4 would not follow, and the Poisson convergence argument would collapse. No internal inconsistency was found; this is a dependency concern about an external result.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the K-symmetric exclusion process (K-SEP) on Z started from deterministic or random step profiles with no particles to the right of the origin, and proves that the rescaled point process of particle positions converges in distribution to a Poisson random measure with an explicit exponential intensity. Theorem 3.4 treats the full step profile and gives the limit PRM(Kσλ); Theorem 3.5 treats truncated L-step profiles and identifies three regimes according to the limit of L(t)/b_t, with intensity cνσ(1−e^{−ψ/σ})λ or cνλ; Corollaries 3.7 and 3.8 translate these into joint limits for order statistics and spacings. The proof uses factorial-moment convergence, a semigroup monotonicity comparison imported from [11], and single-random-walk estimates from [6].","tokens_in":32041,"tokens_out":16010,"duration_ms":195821,"significance":"If correct, the paper is a substantial contribution: it establishes the first Poisson process limits for extremes in a locally interacting symmetric particle system with K>1, extends the K=1 results of [6], and gives explicit, parameter-free intensity constants that depend only on the model inputs σ, K, cν, and ψ. The proof strategy is coherent and refreshingly direct: factorial moments are controlled by explicit error bounds built from a semigroup comparison rather than from the strong Rayleigh property used for SEP. The external inputs from [11] and [6] are clearly identified, and the central derivation is internally consistent. The paper also yields new consequences for the joint law of extremes and spacings, which are stated as immediate corollaries of the Poisson limit.","major_comments":[],"minor_comments":[{"comment":"Lemma 5.1 is the single load-bearing external input, so the manuscript should record the verification that the functions to which it is applied are admissible. For the indicator 1_{A^n} used in Corollary 5.2 and Corollary 5.7, the verification is immediate: the quadratic form associated with 1_{x,y∈A} is (Σ_{x∈A} β(x))^2 ≥ 0 for all sum-zero β, and the n-variable indicator is positive definite in each pair of variables. Adding this one-sentence check would remove the only dependency concern in the proof.","section":"Section 5, Lemma 5.1 and Corollary 5.2"},{"comment":"The last inequality in the proof of Lemma 6.6 appears to give a different constant: for y > b/2 one has y² ≤ 4y⁴/b², which yields 16M₄t/b² rather than the displayed 8M₄t/b². In the applications the term vanishes under (2.11) or (2.14), so the asymptotic results are unaffected, but the statement as written is not justified by the argument and should be corrected or strengthened.","section":"Lemma 6.6"},{"comment":"The condition 'If lim_{t→∞} L (log t / t)^{1/2} > 0' is slightly ambiguous because Theorem 3.5 assumes the limit exists; the corollary should state 'if the limit exists and is positive' or use liminf, so that the statement aligns with the hypotheses of Theorem 3.5.","section":"Corollary 3.8"},{"comment":"Please correct typographical errors, including 'extention' in the Introduction, 'Propostion' in the proof of Theorem 3.5(b), 'with with' immediately after (4.17), and the display '1_An' in the proof of Corollary 5.2 (should be 1_{A^n}).","section":"Throughout"},{"comment":"Reference [23] lists the page range '461–415', which appears to be reversed and should be corrected.","section":"References"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is well within the journal's scope and the central claim is sound. The only substantive request is to record the hypothesis check for the imported semigroup inequality in Section 5; the remaining items are local corrections. I do not see a reason to delay acceptance beyond a minor revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper proves a Poisson random measure limit for the whole extremal cloud in K-symmetric exclusion, and that is genuinely new: even for K=1 only the Gumbel maximum was known, and for K>=2 no analogous limit existed. The result is stronger than a single-particle maximum and the three scaling regimes for truncated step profiles are natural and cleanly stated. The intensity constants are explicit functions of the model inputs, so there is no fitted-parameter smell and no circularity.\n\nWhat the paper does well: the proof strategy is coherent and economical. It uses factorial moments, a semigroup comparison in place of the strong Rayleigh property, and imports sharp single-random-walk estimates. Proposition 4.3, where the semigroup comparison converts correlation errors into the quantities kappa_t and tau_t, is elegant. The authors are also careful about the weak-convergence criteria: the ring U and the factorial moment lemmas are set up properly, and the order-statistic corollaries follow by routine continuous mapping.\n\nThe main soft spot is exactly the one the stress-test flags: Lemma 5.1 from [11] is load-bearing, and its applicability to the functions used here is not checked. The lemma is stated for symmetric positive definite f on Omega_K^n, but positive definiteness is only defined on Z^n, and in Corollary 5.2 the lemma is applied to 1_{A^n} with A a finite interval or half-line. That application is asserted, not verified. If the [11] theorem requires finite-support or cylinder-type functions, then the bounds in Corollary 5.7 and hence in Propositions 4.3 and 4.5 would need additional argument. I do not think the main theorem is wrong; the inequality for indicators is plausible and the surrounding proof logic is sound. But a referee should ask the authors to spell out why Lemma 5.1 covers their indicators, including the unbounded half-line case. This is a genuine gap, not a manufactured one, and it is proportional: the rest of the proof stands or falls on this one external input.\n\nThe other dependencies, mostly the single-walk estimates from [6], are external but they are published results and the authors cite them cleanly. Self-citation here is not a red flag; the cited items are benchmarks, not the new claim.\n\nWho this is for: probabilists working on interacting particle systems, extreme value theory, and semigroup inequalities. It deserves a serious referee: I would send it out, with an explicit request to verify the applicability of Lemma 5.1. I would also bring it to a reading group; the theorem is worth knowing even while that one lemma is being checked.","headline":"Genuinely new PRM limit for K-SEP extremes; the proof is well-structured and likely correct, with one external semigroup lemma that needs a hypothesis check before the argument is airtight.","tokens_in":32545,"tokens_out":2049,"would_cite":true,"duration_ms":29329,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","60G55","60F05","82C22"],"pacs":[],"model":"deepseek-v4-flash","headline":"For K-symmetric exclusion with up to K particles per site, the rescaled positions of all extreme particles converge to a Poisson random measure with exponential intensity.","keywords":["K-symmetric exclusion","Poisson random measure","extreme value theory","order statistics","Gumbel limit","factorial moments","semigroup monotonicity","interacting particle systems"],"falsifier":"Simulate nearest-neighbor K-SEP with $K=2$ (jump rate 1/2 to each neighbor, so $\\sigma=1$) starting from the full step, and for a fixed bounded interval $A$ measure the empirical second factorial moment of the rescaled point process $N_{t/K}(v_t^{-1}(A))$ at large $t$; Theorem 3.4 predicts it approaches $(2\\lambda(A))^2$. A persistent discrepancy as $t$ grows would refute the Poisson convergence.","tokens_in":31570,"feed_emoji":"🎲","tokens_out":9763,"duration_ms":108854,"temperature":0.7,"pith_summary":"What happens to the far-right particles in a symmetric exclusion system when each site may hold up to $K$ particles? This paper proves that, starting from a step profile in which $K$ particles occupy every nonpositive integer, the entire rescaled cloud of extreme particles converges to a Poisson random measure with intensity $K\\sigma e^{-x}\\,dx$, where $\\sigma$ is the jump standard deviation. The rescaling is superdiffusive: positions are divided by $\\sqrt{t/\\log t}$ and shifted by $\\log\\left(t/(\\sqrt{2\\pi}\\log t)\\right)$, with time sped up by $K^{-1}$ to normalize jump rates. The result is stronger than the previously known Gumbel law for the single maximum, because it fixes the joint law of all order statistics and their spacings; it is new even for the classical $K=1$ exclusion process. The paper also analyses finite step profiles supported on blocks of length $L(t)$, obtaining Poisson limits in three regimes determined by whether $L(t)$ is much larger, comparable to, or much smaller than $\\sqrt{t/\\log t}$. This matters because it suggests the extremal Poisson statistics are a general feature of symmetric local interactions, not an accident of independent particles or of the strong negative-correlation structure used in earlier SEP proofs.","feed_headline":"K-symmetric exclusion extremes converge to a Poisson process","feed_subtitle":"Superdiffusive rescaling turns the whole tail of particles into a Poisson random measure, pinning down joint laws and spacings.","key_machinery":"Three ingredients carry the argument. The stirring construction represents K-SEP by K labeled random walks per site that swap positions at Poisson clocks; the set of particle positions equals the set of stirring positions, and each marginal trajectory is a rate-$K$ random walk, so time rescaling by $K^{-1}$ standardizes rates. The factorial moment method then reduces Poisson convergence to showing that factorial moments of counts on finite unions of intervals converge to powers of the limiting intensity (Lemmas 2.1-2.4). The load-bearing correlation estimate is the semigroup monotonicity inequality imported from [11]: for any symmetric positive definite function $f$, the $n$-particle K-SEP semigroup $V_K^n(t)f$ is bounded above by the independent-motion semigroup $U_K^n(t)f$. Combined with sharp single-random-walk tail and local central limit bounds, this controls the error terms $\\kappa_t$ and $\\tau_t$ and forces them to vanish in each scaling regime.","core_discovery":"Theorem 3.4 states: under symmetric, irreducible, translation-invariant jump rates with finite exponential moments, $N_{t/K}\\circ v_t^{-1} = \\sum_{m=0}^\\infty \\delta_{v_t(X^{(m)}_{t/K})}$ converges in distribution to $\\mathrm{PRM}(K\\sigma \\lambda)$ with $\\lambda(dx)=e^{-x}\\,dx$. Theorem 3.5 extends this to product step initial conditions with average density $c_\\nu$: for truncated blocks of length $L(t)$ with $L(t)\\sqrt{\\log t/t}\\to\\psi\\in(0,\\infty]$, the limit is $\\mathrm{PRM}(c_\\nu\\sigma(1-e^{-\\psi/\\sigma})\\lambda)$; when $L(t)\\sqrt{\\log t/t}\\to 0$, a different scale $b_{t,L}=(t/\\log L^2)^{1/2}$ yields $\\mathrm{PRM}(c_\\nu\\lambda)$. From these Poisson limits the paper derives the joint convergence of any finite tuple of order statistics to $-\\log(T_m/c)$, with $T_m$ sums of independent exponential random variables, and the convergence of neighboring spacings to independent exponential laws with rates $1,2,3,\\dots$ . These claims make precise the sense in which the extremal cloud decouples into independent exponential gaps.","pith_inferences":["Editorial extension: any symmetric particle system that admits a stirring representation and a semigroup comparison of this type should show the same Poisson $e^{-x}$ extremal limits; the strong Rayleigh property is not essential.","Editorial extension: as $K$ grows, K-SEP on the $K^{-1}$ time scale approaches independent motion, so the intensity $K\\sigma\\lambda$ should agree with the extremal process of independent random walks starting with $K$ particles per site; checking this consistency would test both limits at once.","Editorial extension: the block-length transition near $L\\sim b_t$ suggests a critical window in which the limiting intensity constant varies continuously between the two regimes; deriving an explicit interpolation formula at the critical scale is a natural next step."],"forward_implications":["The maximum particle $X^{(0)}_{t/K}$, rescaled by $v_t$, converges to a Gumbel law with location parameter determined by $c_\\nu\\sigma(1-e^{-\\psi/\\sigma})$ (or by $c_\\nu$ in the short-block regime), extending the $K=1$ result to all $K\\ge 1$.","Every finite tuple of order statistics converges jointly to the transformed exponential sums $-\\log(T_m/c)$, so the joint extremal law, not just the marginal maximum, is characterized.","Spacings between consecutive extremes, divided by $\\sigma b_t$, converge to independent exponential variables with rates $1,2,3,\\ldots$, and the limiting spacing law is independent of the initial condition.","Particles starting farther than the scale $b_t$ behind the origin do not influence the limiting extreme cloud; only the average density $c_\\nu$ near the origin survives in the intensity.","The point-process limit is new even for $K=1$ SEP, going beyond the previously known Gumbel marginal by describing the entire extremal cloud."],"supporting_citations":[{"why":"Supplies the sharp single-random-walk tail and local central limit estimates used to make the factorial-moment errors vanish, and the SEP Gumbel result being extended.","marker":"[6]"},{"why":"Gives the semigroup monotonicity inequality $V_K^n(t)f\\le U_K^n(t)f$ for symmetric positive definite $f$ that controls K-SEP correlations.","marker":"[11]"},{"why":"Provides the criteria that turn factorial-moment convergence into Poisson random measure convergence in distribution, plus the thinning lemma used for product initial conditions.","marker":"[13]"},{"why":"Gives the $K=1$ semigroup comparison and integration-by-parts identity adapted in Section 5 to the K-SEP semigroup.","marker":"[18]"},{"why":"Provides the independent-random-walk order-statistic and spacing limits that the K-SEP limits generalise and are compared with.","marker":"[20]"}],"fun_headline_variants":["K-symmetric exclusion: tail extremes converge to Poisson","Extremes in K-exclusion become Poisson under superdiffusive scaling","Poisson limit for extremes in K-symmetric exclusion processes","K-exclusion: order statistics converge to Poisson point process","Tail particles in K-symmetric exclusion form a Poisson cloud"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof stands on the imported semigroup monotonicity inequality that bounds K-SEP correlations by independent-motion correlations; if that inequality failed, the factorial-moment error estimates would not vanish and the Poisson limit would not follow.","fun_headline_variants_meta":{"raw":{"variants":["K-symmetric exclusion: tail extremes converge to Poisson","Extremes in K-exclusion become Poisson under superdiffusive scaling","Poisson limit for extremes in K-symmetric exclusion processes","K-exclusion: order statistics converge to Poisson point process","Tail particles in K-symmetric exclusion form a Poisson cloud"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000201,"raw_usage":{"total_tokens":1505,"prompt_tokens":1195,"completion_tokens":310,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":811,"completion_tokens_details":{"reasoning_tokens":227}},"tokens_in":811,"tokens_out":310,"duration_ms":3871,"temperature":1.0,"reasoning_tokens":227,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T00:45:59.602901+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate nearest-neighbor K-SEP with $K=2$ (jump rate 1/2 to each neighbor, so $\\sigma=1$) starting from the full step, and for a fixed bounded interval $A$ measure the empirical second factorial moment of the rescaled point process $N_{t/K}(v_t^{-1}(A))$ at large $t$; Theorem 3.4 predicts it approaches $(2\\lambda(A))^2$. A persistent discrepancy as $t$ grows would refute the Poisson convergence.","supporting_citations":[{"cited_title":"Conroy and S","cited_arxiv_id":null,"evidence_quote":"Supplies the sharp single-random-walk tail and local central limit estimates used to make the factorial-moment errors vanish, and the SEP Gumbel result being extended."},{"cited_title":"Giardin` a, F","cited_arxiv_id":null,"evidence_quote":"Gives the semigroup monotonicity inequality $V_K^n(t)f\\le U_K^n(t)f$ for symmetric positive definite $f$ that controls K-SEP correlations."},{"cited_title":"Kallenberg.Random Measures, Theory and Applications(2017)","cited_arxiv_id":null,"evidence_quote":"Provides the criteria that turn factorial-moment convergence into Poisson random measure convergence in distribution, plus the thinning lemma used for product initial conditions."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the $K=1$ semigroup comparison and integration-by-parts identity adapted in Section 5 to the K-SEP semigroup."},{"cited_title":"Mikosch and J","cited_arxiv_id":null,"evidence_quote":"Provides the independent-random-walk order-statistic and spacing limits that the K-SEP limits generalise and are compared with."}],"review_version":1}