{"id":"1217c090-1a04-40f8-9cbc-d50ccfa638f4","arxiv_id":"2501.10522","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For symmetric exclusion on Z^d, d>=2, with step and polynomial-shape initial conditions, particle counts beyond a suitable superdiffusive level are asymptotically Poisson, implying Gumbel limits for the rightmost particle and all order statistics.","lead":"For a many-particle system with exclusion interactions on a high-dimensional lattice, this paper proves that when particles from a half-space leak into the empty side, the number crossing a far-away line becomes Poisson-distributed exactly as if particles ignored each other. This yields the first extreme-value (Gumbel) limit laws for the exclusion process in dimensions greater than one and clarifies when particle correlations can be ignored at extreme scales.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Condition (4.2) in d=2,3 is an unproven extra hypothesis; the abstract's claim that convergence of E[N_t] alone suffices overstates low-dimensional generality, though Theorem 5.1's polynomial case is unaffected.","rationale":"The reader's CONDITIONAL verdict is well calibrated. I re-read the proof chain for Theorem 5.1 and found no computational error in the polynomial case: Lemma 5.3 evaluates the mean via Lemmas 9.4/9.5, and Lemma 5.4 supplies the St/Ct bounds (5.10)-(5.12), which are more than enough to drive E_t→0 and also imply (4.2) for d=2,3. The d=2,3 extra hypothesis in Theorem 4.1 is therefore not a defect of the main application. It is, however, a real limitation of the abstract's headline claim. The paper itself (Section 2.4, Remark 4.4) identifies necessity of (4.2) as open, so the concern is disclosed; still, the abstract's unconditional 'when lim E[N_t] exists' phrasing is stronger than what is proved. Equation (7.1) is a typo, not load-bearing. Accordingly, the reader's CONDITIONAL verdict should stand without further adjustment.","tokens_in":36750,"tokens_out":28126,"duration_ms":277146,"concrete_test":"For d=3, take a shape satisfying Conditions (A)-(C) with G(u)=exp((log u)^2), for example g_2(u) = exp((log u)^2)/3, so g_2'/g_2 → 0. Choose z(t) so that E0[∫_0^{(ζ_{t/3}-z)_+} G(u) du] has a positive finite limit, using the Gaussian reduction of Lemmas 9.4-9.5 to determine the leading order of z. Then compute r(t)=E0[G(ζ_{t/3}-z)1(ζ_{t/3}>z)] by the same Gaussian asymptotics or by direct random-walk Monte Carlo for t=10^4, 10^6, 10^8. If r(t)(log t)^{1/2} → ∞, then (4.2) fails while E[N_t] converges, showing the abstract's unconditional low-dimensional statement is false. If it does not fail, repeat with G(u)=exp((log u)^γ) for increasing γ>2 until the threshold is crossed; whichever value of γ first violates (4.2) settles the scope question.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing concern is the status of condition (4.2) in dimensions 2 and 3. Theorem 4.1 and Corollary 4.3 are the engine of the paper, and in d=2,3 they require E0[G(ζ_{t/d}-z)1(ζ_{t/d}>z)] = o((log t)^{-1/2}) for d=3 or o(t^{-1/4}) for d=2, in addition to convergence of E[N_t]. The abstract's opening claim sounds unconditional: 'when lim E[N_t] exists, correlations ... vanish so as to allow convergence of N_t to the same Poisson distribution.' For d=2,3 this is only proved under (4.2), and the paper itself leaves necessity of (4.2) open in Section 2.4. Thus the low-dimensional generality advertised in the abstract is not established. This does not threaten the polynomial-shape result Theorem 5.1: Lemma 5.4's proof supplies the stronger bound E0[G(ζ_{t/d}-z)1(ζ_{t/d}>z)] ≤ C e^{-βx}(log t/t)^{1/2}, which implies (4.2) for both d=2 and d=3. The concern is instead about the stated scope of the main theorem outside the polynomial class, where subexponential shapes satisfying (A)-(C) could in principle make the left side of (4.2) decay too slowly while E[N_t] converges. Equation (7.1) also contains a typo ('=0' should be '→0') but that is cosmetic compared with this scope issue.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the symmetric simple exclusion process (SSEP) on Z^d, d ≥ 2, starting from half-space step initial conditions whose cross-section is determined by shape functions g_2,...,g_d. It defines N_t as the number of particles that have moved beyond a level z = z(t) in the first coordinate, and asks whether convergence of E[N_t] suffices for a Poisson limit, as in the corresponding system of independent particles. The main theorem (Theorem 4.1) shows that, under conditions (A)-(C) on the shape functions and uniform boundedness of E[N_t], the correlation quantity E_t = S_t + C_t tends to 0: unconditionally for d ≥ 4, and in d = 2,3 under an additional decay condition (4.2). Corollary 4.3 converts this into a Poisson limit for N_t when lim E[N_t] exists. For polynomial shapes g_i(u) = c_i u^{α_i} + r_i, Theorem 5.1 identifies the scaling z = b_t(x+a_t), proves N_t converges to a Poisson law with mean M e^{-βx}, and consequently obtains a Gumbel limit for the maximum and explicit finite-sum limits for all order statistics. The proofs are built on the stirring construction, self-duality, negative association and the strong Rayleigh property, and are carried out in Sections 6-9 with auxiliary lemmas stated in detail.","tokens_in":37143,"tokens_out":11482,"duration_ms":110821,"significance":"If correct, this is a substantial contribution: it provides the first extreme-value and Poisson-statistics limits for the exclusion process in dimensions d > 1, showing that the statistics of the particle front match those of independent particles for a nontrivial class of interacting initial conditions. The d ≥ 4 result is an unconditional statement under natural hypotheses, and the polynomial-shape theorem gives explicit scaling and dimension-dependent constants that are directly checkable. A particular strength is that the proofs are largely self-contained and include quantitative rates of decay for S_t and C_t in Lemma 5.4; no free parameters are fitted and the main predictions are sharp and falsifiable. The d = 2,3 part is correctly proved only under the additional condition (4.2), whose necessity is explicitly left open; this qualification must be reflected in the abstract and introductory claims.","major_comments":[{"comment":"The abstract states without qualification that 'when lim_{t→∞} E[N_t] exists, correlations between particles beyond z vanish so as to allow convergence of N_t to the same Poisson distribution' as for independent particles. This is stronger than what is proved. In d = 2,3, Theorem 4.1 and Corollary 4.3 require the additional condition (4.2) on E_0[G(ζ_{t/d}-z)1(ζ_{t/d}>z)], and the necessity of (4.2) is left open in Section 2.4 and Remark 4.4(a). The paper should qualify the abstract and the corresponding sentences in the Introduction and Section 2.2, either by stating that in d = 2,3 the result is conditional on (4.2) or by restricting the unqualified statement to d ≥ 4. This is a scope correction rather than a mathematical error: Theorem 5.1 is unaffected because Lemma 5.4 verifies (4.2) for polynomial shapes, but the advertised generality of the main theorem needs to be corrected before publication.","section":"Abstract, Theorem 4.1, Eq. (4.2), Section 2.4"}],"minor_comments":[{"comment":"The display preceding the proof of Theorem 4.1 reads γ_d(t)(E_0[G(ζ_{t/d}-z)1(ζ_{t/d}>z)])^2 = 0 for d ∈ {2,3}; the intended statement is that this quantity tends to 0 as t → ∞. As written, the equality makes the subsequent argument in (7.2) meaningless. Please correct '= 0' to '→ 0'.","section":"Section 7, Eq. (7.1)"},{"comment":"The displayed Gumbel rate appears to have (d-1)! ∏_i(2c_i) d^{d+1/2} e^{-dx} in the exponential. Theorem 5.1 and Remark 5.2(c) give the same rate with d^{d+1/2} in the denominator, i.e. (d-1)!∏_i(2c_i) d^{-(d+1/2)} e^{-dx}. If the inline formula is not a typesetting artifact, the exponent should be corrected.","section":"Section 2.2, Eq. (2.4)"},{"comment":"In the definition of \\anH^i_{k,s}(m), the product over l ≠ i contains ζ_{u/d} with an undefined variable u; it should be ζ_{s/d} to agree with the rest of the lemma. A similar u/s typo appears in the proof of (8.17), where one expectation uses ζ_{u/d} instead of ζ_{s/d}.","section":"Section 8, Lemma 8.2 and its proof"}],"recommendation":"minor_revision","confidential_remarks":"This is a solid paper with a correct main proof. The only substantive issue is the abstract's unqualified statement of the d = 2,3 result, which should be corrected to reflect the conditional status of Theorem 4.1 in low dimensions. The remaining issues are local typos. The heavy self-citation of [4] is justified by the shared proof template and does not create circularity. I would be happy to see the paper accepted after the requested qualifications and typographical corrections are made."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: this is a real advance over the d=1 paper by the same authors, with full proofs, and the Gumbel/order-statistic limits for polynomial shapes in d≥2 look correct. The only substantive complaint is scope: the abstract promises more than Theorem 4.1 delivers in d=2,3.\n\nWhat is genuinely new: the multidimensional step-profile formalism, the correlation bounds in Proposition 4.6 that decay at a dimension-dependent rate through gamma_d(t), and the explicit limit (5.3) with the constant M in (5.4) for polynomial shaping functions. The proof of Theorem 5.1 is complete: Lemma 5.4 gives the sharper bounds needed to verify condition (4.2) in d=2,3, so the Gumbel result is not hostage to the extra hypothesis.\n\nThe soft spot is exactly the one the stress-test note flags. In d=2,3, Theorem 4.1 needs condition (4.2) on top of convergence of E[N_t]. The abstract says 'in large generality' that convergence of E[N_t] makes correlations vanish, which is only established for d≥4 unconditionally; in low dimensions it requires an extra geometric hypothesis whose necessity is explicitly left open in Section 2.4. That's an overstatement in the abstract, but the paper itself is honest about it. The typo in (7.1) (the '=0' should be '→0') is cosmetic and obvious from context.\n\nThe citation pattern is fine: the heavy self-citation of [4] is justified because the proof structure extends that paper, and the new d>1 bounds are genuinely new. I did not find circularity or fitted constants. The random walk estimates in the appendix are standard and correctly applied.\n\nWho is this for: anyone working on extremes of interacting particle systems, or on Poisson convergence for weakly dependent fields. It deserves a serious referee. The referee will need to check the long Sections 6-9, but the main ideas are clear. My recommendation: send to peer review, and ask the authors to adjust the abstract's d=2,3 claim and fix the typo.","headline":"Solid extension of the d=1 Gumbel work to d>1 with detailed proofs; the abstract overstates the d=2,3 scope slightly but the polynomial-shape theorem holds.","tokens_in":37642,"tokens_out":2088,"would_cite":true,"duration_ms":19704,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","60F05"],"pacs":[],"model":"deepseek-v4-flash","headline":"For symmetric exclusion in d>1, extremes match independent-particle statistics when the expected count converges, with an explicit Gumbel limit for polynomial initial shapes.","keywords":["symmetric exclusion process","SSEP","step initial condition","Poisson convergence","Gumbel limit","order statistics","particle correlations","random walk"],"falsifier":"In d = 2 or 3, construct a shape function g_i satisfying Conditions (A)–(C) with sup_t E[N_t] < ∞ and lim E[N_t] = λ > 0 but with E_0[G(ζ_{t/d} − z)1(ζ_{t/d} > z)] decaying slower than the rates in (4.2); then simulate or estimate S_t + C_t to see whether it stays bounded away from zero. If it does, the Poisson convergence in Corollary 4.3 fails for that profile, disproving the sufficiency of the mean-convergence condition without the extra hypothesis. A direct numerical check for polynomial shapes in d = 2 and 3, comparing the empirical distribution of N_t to Poisson(M $e^{{−βx}}$) at finite t and verifying the Gumbel tail, would also settle whether the stated rates are sharp.","tokens_in":36531,"feed_emoji":"🎲","tokens_out":3662,"duration_ms":39348,"temperature":0.7,"pith_summary":"This paper studies the symmetric simple exclusion process (SSEP) on Z^d, d ≥ 2, started from half-space 'step' profiles shaped by functions g_i. Its central claim is that, in large generality, if the expected number of particles crossing a moving frontier z(t) has a limit, then particle correlations beyond z vanish and the count N_t converges to the same Poisson law as if particles moved independently. In d ≥ 4 this follows from convergence of the mean alone; in d = 2, 3 an extra geometric condition on the profile is needed. For polynomial profiles the paper identifies the correct scaling explicitly and derives a Gumbel limit for the maximum particle position, plus limits for all order statistics. If true, this means interactions do not alter the extreme-value statistics of exclusion in high dimensions.","feed_headline":"Exclusion extremes match independent particles in d>1","feed_subtitle":"When the expected frontier count converges, correlations vanish and the maximum follows a Gumbel law with explicit scaling.","key_machinery":"The argument runs through the stirring representation of SSEP, self-duality, and negative association. The load-bearing identity is the strong Rayleigh criterion (Lemma 3.8): for each t the sum of occupation variables in any finite region has the same law as a sum of independent Bernoulli variables, so N_t ⇒ Poisson(λ) iff E[N_t] → λ and E[N_t] − Var(N_t) → 0. The correlation term E[N_t] − Var(N_t) = S_t + C_t is then controlled by Lemma 2.1, which bounds C_t by an integral over a two-particle displacement kernel, and by Propositions 4.5–4.6, which reduce the bounds to expectations of the one-dimensional random walk ζ_{t/d}. The dimension enters through γ_d(t), the expected number of returns to 0 of a (d−1)-dimensional random walk: √t for d = 2, log t for d = 3, and 1 for d ≥ 4; this drives the sharpness of correlation decay. For polynomial profiles, Gaussian large-deviation asymptotics (Lemma 9.4) evaluate the mean E[N_t] and the rates of S_t and C_t.","core_discovery":"The paper establishes that for SSEP in d ≥ 2 with step initial conditions satisfying Conditions (A)–(C), the criterion for Poisson convergence of N_t is the same as for independent Bernoulli arrays: lim E[N_t] = λ and lim Var(N_t) = λ, equivalently E[N_t] − Var(N_t) → 0. Using the strong Rayleigh property, this reduces to showing that the quantities S_t (sum of squared single-particle occupation probabilities) and C_t (sum of negative covariances) tend to zero. The main theorem proves S_t + C_t → 0 whenever sup_t E[N_t] < ∞ in d ≥ 4, and in d = 2, 3 under condition (4.2), which bounds E_0[G(ζ_{t/d} − z)1(ζ_{t/d} > z)] by explicit o-rates. For polynomial shapes g_i(u) = c_i $u^{{α_i}}$ + r_i, the paper identifies z = b_t(x + a_t) with b_t = (βt/(d log t))^{1/2} and a_t = log(t / ((2π)^{1/(2β)} (log t)^{(β+1)/(2β)})), β = 1 + Σ α_i, and proves N_t ⇒ Poisson(M $e^{{−βx}}$) with the constant M in (5.4). Consequently, the rescaled maximum satisfies a Gumbel law and each order statistic $X_t^{{(m)}}$ has limit (5.3).","pith_inferences":["The result suggests a universality principle for d > 1: the extreme-value statistics of the exclusion process coincide with those of non-interacting random walks whenever the expected frontier count stabilizes, so interaction effects are asymptotically invisible in the far tail even though they shape the bulk.","The extra condition (4.2) in d = 2, 3 may well be necessary for subexponential but non-polynomial shapes, because the slower decorrelation in low dimensions could leave a residual correlation that no choice of level z can remove; a counterexample would settle the open question the paper raises.","Since the paper's machinery reduces everything to one-dimensional random-walk functionals, one could numerically test convergence rates for finite t by simulating the random walk expectations E_0[G(ζ_{t/d} − z)1(ζ_{t/d} > z)] and comparing S_t + C_t bounds to the predicted Poisson behavior, even without running the full exclusion process.","The explicit Gaussian large-deviation route suggests that the correct frontier for shapes with growth between polynomial and exponential may interpolate between √(t log t) and linear scale t, and the sharp transition point could be identified by analyzing when the tail P(ζ_{t/d} > z) becomes so small that expectations of G(ζ_{t/d} − z) cease to dominate."],"forward_implications":["If the central claim is right, then in d ≥ 4 the Poisson limit for the number of particles beyond a moving frontier follows directly from convergence of its expectation, for every shape satisfying the mild Conditions (A)–(C), with no extra hypothesis.","For polynomial profiles, the explicit scaling gives the maximum position X_t with mean of order √((β/d) t log t) and variance of order (π²/(6β)) t / log t, so extremes move faster than the diffusive scale √t.","The limiting constant M depends only on the leading-order coefficients c_i and the exponents α_i, not on the intercepts r_i, so adding or subtracting finitely many particles (or periodically sparse particles) does not change the extreme-value asymptotics.","The order-statistics limit (5.3) gives the full joint asymptotic distribution of the top particles: for each fixed m, P(X_t^{(m)} ≤ b_t(x+a_t)) converges to the Poisson tail sum, meaning the top particles behave like the record points of an independent-particle system.","In periodic product-measure initial conditions, the same extremes result holds with the mean scaled by the average density ρ̄, so the conclusion is robust to deterministic initial configurations beyond pure step profiles."],"supporting_citations":[{"why":"Supplies the d = 1 Gumbel limit for SSEP and the Lemma 3.4 covariance bound that Lemma 2.1 generalizes; also provides the random-walk large-deviation estimates used in Lemma 9.4.","marker":"[4]"},{"why":"Provides the correlation inequality (Lemma 1' in [1]) that bounds the two-particle hitting probability, a key ingredient in Lemma 3.6 and the covariance estimates.","marker":"[1]"},{"why":"The standard reference for the symmetric exclusion process, giving the stirring construction, negative correlation (Proposition VIII.1.7), and the V_2 ≤ U_2 order used in Lemma 2.1.","marker":"[12]"},{"why":"Establishes the Bernoulli-representation criterion (Lemma 3.8) that reduces Poisson convergence of N_t to convergence of E[N_t] and vanishing of E[N_t] − Var(N_t).","marker":"[14]"},{"why":"Supplies the strong Rayleigh property for product initial conditions, which underpins the Bernoulli representation and the nonnegativity of the covariance terms.","marker":"[3]"},{"why":"Provides the classical large-deviation bound for simple random walks used in Lemma 9.4 to replace random-walk tail probabilities by Gaussian tails at scales z = o(t^{2/3}).","marker":"[7]"},{"why":"Gives the random-walk transition probability bounds (8.4)–(8.5) used throughout Section 8 to estimate the sums over lattice points in the covariance bound.","marker":"[11]"}],"fun_headline_variants":["Poisson frontier for symmetric exclusion in d>1","Vanishing correlations yield Poisson count in exclusion","Gumbel extremes from vanishing correlations","Symmetric exclusion: Poisson frontiers and Gumbel maxima"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"In d = 2 and 3, the conclusion that correlations vanish and N_t becomes Poisson rests on the extra geometric condition (4.2), which the paper verifies for polynomial shapes but does not prove is necessary for all allowed profiles; if this condition fails for some subexponential shape with sup_t E[N_t] < ∞, the main theorem no longer applies in low dimensions.","fun_headline_variants_meta":{"raw":{"variants":["Poisson frontier for symmetric exclusion in d>1","Vanishing correlations yield Poisson count in exclusion","Gumbel extremes from vanishing correlations","Symmetric exclusion: Poisson frontiers and Gumbel maxima"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000771,"raw_usage":{"total_tokens":3466,"prompt_tokens":1046,"completion_tokens":2420,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":662,"completion_tokens_details":{"reasoning_tokens":2361}},"tokens_in":662,"tokens_out":2420,"duration_ms":18938,"temperature":1.0,"reasoning_tokens":2361,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T19:09:18.850279+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"In d = 2 or 3, construct a shape function g_i satisfying Conditions (A)–(C) with sup_t E[N_t] < ∞ and lim E[N_t] = λ > 0 but with E_0[G(ζ_{t/d} − z)1(ζ_{t/d} > z)] decaying slower than the rates in (4.2); then simulate or estimate S_t + C_t to see whether it stays bounded away from zero. If it does, the Poisson convergence in Corollary 4.3 fails for that profile, disproving the sufficiency of the mean-convergence condition without the extra hypothesis. A direct numerical check for polynomial shapes in d = 2 and 3, comparing the empirical distribution of N_t to Poisson(M $e^{{−βx}}$) at finite t and verifying the Gumbel tail, would also settle whether the stated rates are sharp.","supporting_citations":[{"cited_title":"Conroy and S","cited_arxiv_id":null,"evidence_quote":"Supplies the d = 1 Gumbel limit for SSEP and the Lemma 3.4 covariance bound that Lemma 2.1 generalizes; also provides the random-walk large-deviation estimates used in Lemma 9.4."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the correlation inequality (Lemma 1' in [1]) that bounds the two-particle hitting probability, a key ingredient in Lemma 3.6 and the covariance estimates."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The standard reference for the symmetric exclusion process, giving the stirring construction, negative correlation (Proposition VIII.1.7), and the V_2 ≤ U_2 order used in Lemma 2.1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the Bernoulli-representation criterion (Lemma 3.8) that reduces Poisson convergence of N_t to convergence of E[N_t] and vanishing of E[N_t] − Var(N_t)."},{"cited_title":"Borcea, P","cited_arxiv_id":null,"evidence_quote":"Supplies the strong Rayleigh property for product initial conditions, which underpins the Bernoulli representation and the nonnegativity of the covariance terms."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the classical large-deviation bound for simple random walks used in Lemma 9.4 to replace random-walk tail probabilities by Gaussian tails at scales z = o(t^{2/3})."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the random-walk transition probability bounds (8.4)–(8.5) used throughout Section 8 to estimate the sums over lattice points in the covariance bound."}],"review_version":1}