Point process convergence of extremes in $K$-symmetric exclusion

Adri\'an Gonz\'alez Casanova; Michael Conroy; Sunder Sethuraman

Point process convergence of extremes in $K$-symmetric exclusion

T0 review · 0 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2506.12632 v1 pith:33UE3YWE submitted 2025-06-14 math.PR

Michael Conroy , Adrián González Casanova , Sunder Sethuraman This is my paper · ORCID

classification math.PR MSC 60K3560G5560F0582C22

keywords K-symmetricexclusionPoissonrandommeasureextremevaluetheoryorderstatisticsGumbellimitfactorialmomentssemigroupmonotonicityinteractingparticlesystems

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

What 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.

The 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.

The 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.

Who 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.

Referee Report

0 major / 5 minor

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].

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.

minor comments (5)

[Section 5, Lemma 5.1 and Corollary 5.2] 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.
[Lemma 6.6] 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.
[Corollary 3.8] 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.
[Throughout] 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}).
[References] Reference [23] lists the page range '461–415', which appears to be reversed and should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Poisson random measure limits are derived from explicit model inputs, and the cited semigroup inequality and prior K=1 estimates are independent external results.

full rationale

The derivation is self-contained given its cited external inputs. Theorems 3.4 and 3.5 establish Poisson random measure convergence via the factorial-moment criteria (Lemmas 2.3 and 2.4), and all limiting intensities are explicit functions of model data: K, sigma, c_nu, and psi. These intensities are obtained from mean-measure limits in Proposition 4.1, not from fitting any parameter to the target point process. The error estimates depend on the semigroup comparison Lemma 5.1 imported from [11], which is an independent result comparing the K-SEP semigroup to that of independent motion, and on single-random-walk estimates from [6]. Although [6] is prior work of two of the present authors, its cited statements are K=1 mean-convergence and tail/local-CLT estimates whose assumptions do not include the K>=2 point-process limit proved here; they are external, falsifiable results and therefore do not constitute circular self-citation under the stated rules. The applications of Lemma 5.1 to indicator functions in Corollaries 5.2 and 5.7 are made within the lemma's stated positive-definite framework as used; any concern about extending the inequality to unbounded indicator sets would be a correctness or rigor issue, not a circularity. No equation in the paper reduces a predicted quantity to a fitted parameter, and no self-citation is invoked to forbid alternative derivations. Hence the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The proof imports a semigroup comparison theorem, sharp single-random-walk estimates, and the stirring construction from prior literature; none of these are fitted to the target result and they are all standard or previously published.

assumptions (5)

domain assumption Semigroup monotonicity inequality (Lemma 5.1): for symmetric positive definite f on the K-SEP n-particle state space, V_K^n(t) f ≤ U_K^n(t) f.
Quoted from [11] without proof; it is the main tool comparing K-SEP factorial moments to independent-motion moments in Proposition 4.3 and Corollary 5.7.
domain assumption Sharp random-walk estimates (Lemma 6.5 from [6]) for P_x(ζ_s=0)^2 (and a shifted version) integrated against P_x(ζ_{t-s}>z_t)^2, vanishing for z_t ~ c sqrt(t log t) or z_t ~ c sqrt(t log L).
Used in the proof of Proposition 4.4(a) to show the correlation error κ_t tends to zero; cited from the authors' earlier work.
domain assumption Stirring representation of K-SEP (Section 2.1): K labeled continuous-time random walks per site moving at rate K and swapping at rate K^2 p(x,y) produce the generator (2.1) after projection.
Standard graphical construction assumed throughout; it justifies the key representation N_t = Σ_{x,j} 1(η_0(x)≥j) δ_{ξ_t^{xj}} in (2.8).
domain assumption Jump distribution assumptions (Condition 3.1): p symmetric, irreducible, p(0,0)=0, with finite exponential moment near 0.
Defines the model class; used for the CLT tail bounds and superdiffusive scaling.
domain assumption Initial condition condition (Condition 3.2): ν is a product measure on the nonpositive integers with positive Cesàro limit c_ν of site means.
Determines the intensity proportionality constant c_ν in Theorem 3.5 and guarantees infinitely many particles.

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Pith. "Pith review of Point process convergence of extremes in $K$-symmetric exclusion." pith.science (2026). https://pith.science/paper/33UE3YWE

@misc{pith2026250612632,
  author       = {Pith},
  title        = {Pith review of: Point process convergence of extremes in $K$-symmetric exclusion},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/33UE3YWE}},
  note         = {Machine review of arXiv:2506.12632}
}

abstract

We consider the behavior of extremal particles in $K$-symmetric exclusion on $\mathbb{Z}$ when the process starts from certain infinite-particle step configurations where there are no particles to the right of a maximal one. In such a system, the occupancy of a site is limited to at most $K \geq 1$. Let $X^{(0)}_t\geq X^{(1)}_t\geq \cdots$ denote the order statistics of the particles in the system. We show that the point process $\sum_{m=0}^\infty \delta_{v_t(X_{t/K}^{(m)})}$ converges in distribution as $t \to \infty$ to a Poisson random measure on $\mathbb{R}$ with intensity proportional to $e^{-x}\,dx$, where $v_t(x) = (\sigma b_t)^{-1}x - a_t$, $a_t = \log(t/ (\sqrt{2\pi} \log t))$, $b_t = (t/\log t)^{1/2}$, and $\sigma$ is the standard deviation of the random walk jump probabilities. From this limit, we further deduce the asymptotic joint distributions for the extreme statistics and the spacings between them. Moreover, to probe effects of the number of particles on the behavior of the extremes, we consider an array of truncated step profiles supported on blocks of $L(t)$ sites at times $t\geq 0$. Letting $L(t) \to \infty$ with $t \to \infty$, we obtain Poisson random measure limits in different scaling regimes determined by $L(t)$. These results show robustness of both previously known and newly introduced superdiffusive scaling limits for the extremes in the symmetric exclusion process ($K=1$) by extending them to the larger class of $K\geq 2$ exclusion. Furthermore, proofs are more general than previously known techniques, relying on moment bounds and a semigroup monotonicity estimate to control particle correlations.

Figures

Figures reproduced from arXiv: 2506.12632 by the authors.

**Figure 1.** We observe two consecutive particle movements over time on the nearest-neighbor ladder graph with K = 4. In the first step, a particle shifts one position to the right and lands in an available spot. In the second step, particle 2 attempts to move to the position occupied by 4, but the exclusion rule prevents its motion, as particles cannot move into already occupied sites. In contrast, under the stirring coupling, … view at source ↗

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Works this paper leans on

26 extracted references · 26 canonical work pages

[11]

Giardin` a, F

C. Giardin` a, F. Redig, and K. Vafayi. Correlation Inequalities for Interacting Particle Systems with Duality.J. Stat. Phys.(2010) 141: 242–263

work page 2010
[6]

Conroy and S

M. Conroy and S. Sethuraman. Gumbel laws in the symmetric exclusion process.Comm. Math. Physics(2023) 402: 723–764

work page 2023
[1]

R. Arratia. The motion of a tagged particle in the simple symmetric exclusion system on Z 1.Ann. Probab.(1983) 11(2): 362–373

work page 1983
[2]

Ayyer, S

A. Ayyer, S. Misra. An exactly solvable asymmetric K-exclusion process.J. Phys. A: Math. Theor.(2024) 57: 315001 37

work page 2024
[3]

Borcea, P

J. Borcea, P. Br¨ and´ en, and T. M. Liggett. Negative dependence and the geometry of polynomials.J. Am. Math. Soc.(2009) 22(2): 521–567

work page 2009
[4]

P. Caputo. Spectral gap inequalities in product spaces with conservation laws. InStochas- tic analysis on large scale interacting systems(2004) 39: 53–89, Mathematical Society of Japan

work page 2004
[5]

Cocozza-Thivent

C. Cocozza-Thivent. Processus des Misanthropes.Z. Wahrsch. Verw. Gebiete(1985) 70: 509–523

work page 1985
[7]

Poisson statistics, vanishing correlations, and extremal particle limits for symmetric exclusion in d > 1

M. Conroy and S. Sethuraman. Poisson statistics, vanishing correlations, and extremal particle limits for symmetric exclusion ind >1.Preprint, arXiv:2501.10522v2(2024)

work page Pith review arXiv 2024

Show all 26 references

[8]

Floreani, F

S. Floreani, F. Redig, F. Sau. Hydrodynamics for the partial exclusion process in random environment.Stoch. Proc. Appl.(2021) 142: 124–158

2021
[9]

Franceschini, P

C. Franceschini, P. Gon¸ calves, B. Salvador. Hydrodynamical behavior for the symmetric simple partial exclusion with open boundary.Math. Phys., Anal., Geom.(2023) 26:11

2023
[10]

Giardin` a, Jorge Kurchan, F

C. Giardin` a, Jorge Kurchan, F. Redig, and K. Vafayi. Duality and hidden symmetries in interacting particle systems.J. Stat. Phys.(2009) 135: 25–55

2009
[12]

Gon¸ calves, K

P. Gon¸ calves, K. Hayashi, M. Sasada. Characterization of gradient condition for asym- metric partial exclusion processes and their scaling limits.ALEA Lat. Am. J. Probab. Math. Stat.(2025) 22(1): 331–356

2025
[13]

Kallenberg.Random Measures, Theory and Applications(2017)

O. Kallenberg.Random Measures, Theory and Applications(2017). Springer Series in Probability Theory and Stochastic Modelling, vol 77. Springer, Cham

2017
[14]

J. D. Kiesling. An ergodic theorem for the symmetric generalized exclusion process. Markov. Proc. Rel. Fields(1998) 4:351–379

1998
[15]

Kipnis, C

C. Kipnis, C. Landim, and S. Olla. Hydrodynamical limit for nongradient systems: the generalized symmetric exclusion process.Comm. Pure Appl. Math.(1994) 47(11): 1475– 1545

1994
[16]

Kolliopoulos, M

N. Kolliopoulos, M. Larsson, and Z. Zhang. Propagation of chaos for maxima of particle systems with mean-field drift interaction.Probab. Theory Relat. Fields(2023) 187: 1093– 1127

2023
[17]

Kolliopoulos, M

N. Kolliopoulos, M. Larsson, and Z. Zhang. Propagation of chaos for point processes induced by particle systems with mean-field drift interaction.J. Theoret. Probab.(2025) 38:27. 38

2025
[18]

T. M. Liggett.Interacting Particle Systems(2005). Springer, Berlin

2005
[19]

T. M. Liggett. Distributional limits for the symmetric exclusion process.Stoch. Process. Appl.(2009) 119: 1–15

2009
[20]

Mikosch and J

T. Mikosch and J. Yslas. Gumbel and Frechet convergence of the maxima of independent random walks.Adv. Appl. Prob.(2020) 52: 213–236

2020
[21]

S. I. Resnick,Extreme Values, Regular Variation and Point Processes. Springer Series in Operations Research and Financial Engineering (2008). Springer, New York. Reprint of the 1987 original

2008
[22]

Sarantsev and L.-C

A. Sarantsev and L.-C. Tsai. Stationary gap distributions for infinite systems of com- peting Brownian particles.Electron. J. Probab.(2017) 22(56): 1–20

2017
[23]

Sepp¨ al¨ ainen

T. Sepp¨ al¨ ainen. Existence of hydrodynamics for the totally asymmetric simpleK- exclusion process.Ann. Probab.(1999) 27:461–415

1999
[24]

E. Tsegaye. Mixing time and cutoff for the K-SEP.Preprint, arXiv:2401.07999(2024)

2024 arXiv
[25]

Vandenberg-Rodes

A. Vandenberg-Rodes. A limit theorem for particle current in the symmetric exclusion process.Electron. Commun. Probab.(2010) 15: 240–252

2010
[26]

X. Xue. Stationary fluctuation for the occupation time of the multi-species stirring process.Preprint, arXiv:2503.06502(2025). 39

2025 arXiv

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [11]

Giardin` a, F

C. Giardin` a, F. Redig, and K. Vafayi. Correlation Inequalities for Interacting Particle Systems with Duality.J. Stat. Phys.(2010) 141: 242–263

work page 2010

[2] [6]

Conroy and S

M. Conroy and S. Sethuraman. Gumbel laws in the symmetric exclusion process.Comm. Math. Physics(2023) 402: 723–764

work page 2023

[3] [1]

R. Arratia. The motion of a tagged particle in the simple symmetric exclusion system on Z 1.Ann. Probab.(1983) 11(2): 362–373

work page 1983

[4] [2]

Ayyer, S

A. Ayyer, S. Misra. An exactly solvable asymmetric K-exclusion process.J. Phys. A: Math. Theor.(2024) 57: 315001 37

work page 2024

[5] [3]

Borcea, P

J. Borcea, P. Br¨ and´ en, and T. M. Liggett. Negative dependence and the geometry of polynomials.J. Am. Math. Soc.(2009) 22(2): 521–567

work page 2009

[6] [4]

P. Caputo. Spectral gap inequalities in product spaces with conservation laws. InStochas- tic analysis on large scale interacting systems(2004) 39: 53–89, Mathematical Society of Japan

work page 2004

[7] [5]

Cocozza-Thivent

C. Cocozza-Thivent. Processus des Misanthropes.Z. Wahrsch. Verw. Gebiete(1985) 70: 509–523

work page 1985

[8] [7]

Poisson statistics, vanishing correlations, and extremal particle limits for symmetric exclusion in d > 1

M. Conroy and S. Sethuraman. Poisson statistics, vanishing correlations, and extremal particle limits for symmetric exclusion ind >1.Preprint, arXiv:2501.10522v2(2024)

work page Pith review arXiv 2024