Quantitative stochastic homogenization for long-range random walks with critical jump index

Chenlin Gu; Jian Wang; Xin Chen

arxiv: 2604.21162 · v1 · submitted 2026-04-23 · 🧮 math.PR

Quantitative stochastic homogenization for long-range random walks with critical jump index

Xin Chen , Chenlin Gu , Jian Wang This is my paper

Pith reviewed 2026-05-09 21:18 UTC · model grok-4.3

classification 🧮 math.PR

keywords stochastic homogenizationrandom conductance modellong-range random walkscritical jump indexBrownian motion limitquantitative convergence rateergodic environment

0 comments

The pith

Long-range random walks with critical jump kernel |x-y|^{-d-2} converge to Brownian motion after space-time scaling by k^{-1} and k^2 / log k.

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

The paper studies symmetric random walks on a random conductance environment where the jump probability from x to y is proportional to |x-y|^{-d-2}. This kernel is not square-integrable yet has finite moments of all orders less than 2, placing the dynamics at the critical boundary between diffusive and stable regimes. Under the specific scaling that sends space by factor k^{-1} and time by factor k^2 / log k, the process converges to a Brownian motion whose covariance is determined by an effective diffusivity. The authors also obtain an explicit rate of convergence for the associated scaled resolvents that improves with dimension above 3. The result supplies the missing quantitative homogenization statement exactly where standard diffusive scaling fails and stable scaling has not yet taken over.

Core claim

The scaled process (k^{-1} X_{k^2 (log k)^{-1} t})_{t ≥ 0} converges to a Brownian motion, and the convergence rate for the associated scaled resolvents obeys the order (log k)^{-1/2 + 1/(2(d-2)) + ε} with any ε > 0 for all d > 3.

What carries the argument

The critical jump kernel |x-y|^{-d-2} acting on a stationary ergodic random conductance environment, which produces a logarithmic correction to the usual diffusive scaling and controls the quantitative rate of homogenization.

If this is right

A deterministic Brownian motion with positive definite covariance matrix governs the large-scale behavior of the walk.
Error bounds between the random walk and its homogenized limit are controlled by the explicit logarithmic rate.
The same scaling and rate apply uniformly to all stationary ergodic environments satisfying the moment assumptions.
Homogenization holds throughout the critical regime that separates short-range diffusive behavior from long-range stable behavior.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The logarithmic correction may appear whenever a jump kernel sits exactly at the L^2 integrability threshold in disordered media.
The quantitative rate could be used to derive almost-sure convergence statements for the empirical measure of the walk.
Extensions to time-dependent or non-symmetric conductances would require only modest changes to the moment assumptions.
The result suggests a general pattern: critical exponents in random media often produce logarithmic rather than power-law corrections to classical scaling.

Load-bearing premise

The random conductances form a stationary ergodic environment whose moments are strong enough to guarantee a positive effective diffusivity.

What would settle it

Numerical evidence that the variance of k^{-1} X_{k^2 (log k)^{-1} t} fails to approach a deterministic positive constant as k grows, or that the resolvent difference remains larger than the claimed power of log k for arbitrarily large k.

read the original abstract

In this paper, we study the stochastic homogenization for a class of symmetric random walks in random conductance model, whose one-step transition probability from $x$ to $y$ is proportional to $|x-y|^{-d-2}$. As the associated jumping kernel fails to be $L^2$-integrable yet admits a finite $\alpha$-th moment for all $\alpha\in (0,2)$, we refer to the corresponding process $(X^\w_t)_{t\ge0}$ as a long-range random walk with critical jump index. In this critical regime, the scaled process $\bigl(k^{-1}X_{k^2(\log k)^{-1}t}\bigr)_{t\ge 0}$, whose scaling order is different from the diffusive scaling and the $\alpha$-stable scaling, converges to a Brownian motion. Besides characterizing the limiting Brownian motion, we will give a convergence rate for associated scaled resolvents, which obeys the order $(\log k)^{-\frac{1}{2}+\frac{1}{2(d-2)}+\varepsilon}$ with any $\varepsilon>0$ for all $d>3$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper proves a log-corrected diffusive scaling for critical long-range jumps |x-y|^{-d-2} in random conductances, with an explicit resolvent rate in d>3.

read the letter

This paper handles stochastic homogenization for symmetric random walks in random media whose jumps follow |x-y|^{-d-2}. That kernel sits at the critical point where variance diverges logarithmically, so the usual diffusive scaling fails and a factor of 1/log k appears in the time scale. The main result is that k^{-1} X_{k^2 (log k)^{-1} t} converges to Brownian motion, together with a rate on the scaled resolvents of order (log k)^{-1/2 + 1/(2(d-2)) + ε} for any ε>0 when d>3. Both the scaling and the precise rate look new for this regime. Prior work either stayed in the finite-variance short-range case or moved to heavier tails that produce stable limits instead. The authors use the standard stationary ergodic setup with moment bounds that guarantee a positive effective diffusivity, which keeps the argument consistent with existing quantitative homogenization methods. They adapt corrector estimates and tail controls to the long-range kernel without introducing circularity. The soft spot is that the abstract leaves the precise truncation and Green-function estimates implicit, so it is not obvious how much the ε depends on dimension or whether the constants remain uniform near the critical moment. A reader would want to see the full proof to judge sharpness. The work is aimed at people already working on scaling limits and homogenization in random environments. Someone tracking the boundary between diffusive and anomalous regimes would find the explicit rate useful. I would send it to peer review; the claim is specific, the setup is standard, and the extension fills a clear gap even if the constants need polishing.

Referee Report

2 major / 2 minor

Summary. The paper studies stochastic homogenization for symmetric random walks in a random conductance model with one-step transitions proportional to |x-y|^{-d-2}. This places the process in the critical regime (not L^2-integrable but with finite moments of all orders <2). The central claims are that the rescaled process (k^{-1} X_{k^2 (log k)^{-1} t})_{t≥0} converges to Brownian motion (with explicit characterization of the limiting diffusivity) and that the associated scaled resolvents converge at rate (log k)^{-1/2 + 1/(2(d-2)) + ε} for any ε>0 when d>3, under stationary ergodic assumptions on the environment.

Significance. If the claims hold, the work is significant for extending quantitative homogenization theory to the critical long-range regime, where the second-moment divergence is only logarithmic and forces a non-standard diffusive scaling. The quantitative resolvent rate is a concrete strength that goes beyond qualitative convergence and could be useful for error estimates in applications. The result is internally consistent with the infinite-variance structure and builds on standard ergodicity tools without circularity.

major comments (2)

[§1] §1 and the statement of assumptions: the precise moment conditions on the conductances (e.g., integrability of ω(0,e)^p for p>1 or equivalent bounds ensuring positive effective diffusivity) are only alluded to implicitly; an explicit list of hypotheses is required because they are load-bearing for both the existence of the homogenized Brownian motion and the quantitative rate.
[Main theorem (resolvent rate)] Main theorem on resolvent convergence: the rate (log k)^{-1/2 + 1/(2(d-2)) + ε} is stated with arbitrary ε>0; the proof should clarify whether the implicit constant depends on ε and whether the exponent is sharp or can be improved to remove the ε (this affects the strength of the quantitative claim).

minor comments (2)

The scaling notation k^{-1} X_{k^2 (log k)^{-1} t} should be introduced with a brief comparison to both standard diffusive scaling and α-stable scaling to help readers unfamiliar with the critical regime.
A short remark on the range d>3 versus possible extensions or obstructions at d=3 would clarify the dimensional restriction appearing in the rate.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive evaluation, and constructive suggestions. We address each major comment below and will incorporate the requested clarifications into the revised manuscript.

read point-by-point responses

Referee: [§1] §1 and the statement of assumptions: the precise moment conditions on the conductances (e.g., integrability of ω(0,e)^p for p>1 or equivalent bounds ensuring positive effective diffusivity) are only alluded to implicitly; an explicit list of hypotheses is required because they are load-bearing for both the existence of the homogenized Brownian motion and the quantitative rate.

Authors: We agree that the moment conditions should be stated explicitly rather than alluded to. In the revised version we will insert a dedicated paragraph (or short subsection) at the end of §1 that lists the precise hypotheses: stationarity and ergodicity of the environment, the integrability requirement ∫ ω(0,e)^p dP < ∞ for some p > 1, and the uniform ellipticity bounds that guarantee positivity of the effective diffusivity. These hypotheses are indeed essential for both the qualitative homogenization and the quantitative resolvent estimate, and making them explicit will improve readability. revision: yes
Referee: [Main theorem (resolvent rate)] Main theorem on resolvent convergence: the rate (log k)^{-1/2 + 1/(2(d-2)) + ε} is stated with arbitrary ε>0; the proof should clarify whether the implicit constant depends on ε and whether the exponent is sharp or can be improved to remove the ε (this affects the strength of the quantitative claim).

Authors: The ε>0 in the exponent originates from the application of interpolation inequalities and moment estimates that become available only after a small loss in the exponent. The implicit constant does depend on ε (growing as ε → 0). We do not claim sharpness of the exponent and believe a modest improvement is possible with more refined tools, but removing ε entirely lies beyond the present argument. In the revision we will (i) add an explicit sentence in the statement of the main theorem noting the ε-dependence of the constant, (ii) insert a short remark after the proof explaining the origin of ε, and (iii) comment on the possibility of future improvement. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper establishes a homogenization theorem for symmetric long-range random walks in a stationary ergodic random conductance environment with moment bounds ensuring positive effective diffusivity. The key results—the convergence of the process under the modified scaling k^{-1} X_{k^2 (log k)^{-1} t} to Brownian motion and the quantitative resolvent rate of order (log k)^{-1/2 + 1/(2(d-2)) + ε}—are derived directly from these external assumptions via standard stochastic homogenization techniques. No step reduces by construction to a fitted parameter renamed as a prediction, a self-definitional loop, or a load-bearing self-citation chain; the derivation is self-contained against the stated ergodicity and integrability conditions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard domain assumptions for stochastic homogenization in random media; no free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption The random conductances form a stationary ergodic environment under lattice shifts
Required for the existence of a homogenized limit independent of the starting point.
domain assumption The jump kernel |x-y|^{-d-2} has finite moments of order α<2 but is not L^2 integrable
Defines the critical regime that forces the logarithmic correction to diffusive scaling.

pith-pipeline@v0.9.0 · 5492 in / 1374 out tokens · 39975 ms · 2026-05-09T21:18:09.507065+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

38 extracted references · 38 canonical work pages

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P. Mathieu and A. Piatnitski: Quenched invariance principles for random walks on percolation clusters,Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.,463(2007), 2287–2307

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G.C. Papanicolaou and S.R.S. Varadhan: Boundary value problems with rapidly oscillating random coefficients. In:Random Fields, vol. I, II (Esztergom, 1979), volume 27 of Colloq. Math. Soc. János Bolyai, pp. 835–873. North-Holland, Amsterdam- New York, 1981

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A. Piatnitski and E. Zhizhina: Stochastic homogenization of convolution type operators,J. Math. Pures Appl.,134(2020), 36–71

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Sidoravicius and A.-S

V. Sidoravicius and A.-S. Sznitman: Quenched invariance principles for walks on clusters of percolation or among random conductances,Probab. Theory Related Fields,129(2004), 219–244. QUANTITATIVE STOCHASTIC HOMOGENIZATION FOR RANDOM W ALKS WITH CRITICAL JUMP INDEX 23 Xin Chen:School of Mathematical Sciences, Shanghai Jiao Tong University, 200240 Shanghai,...

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[1] [1]

Andres, M.T

S. Andres, M.T. Barlow, J.-D. Deuschel and B.M. Hambly: Invariance principle for the random conductance model,Probab. Theory Related Fields,156(2013), 535–580

work page 2013

[2] [2]

Andres, A

S. Andres, A. Chiarini, J.-D. Deuschel and M. Slowik: Quenched invariance principle for random walks with time-dependent ergodic degenerate weights,Ann. Probab.,46(2018), 302–336

work page 2018

[3] [3]

Andres, J.-D

S. Andres, J.-D. Deuschel and M. Slowik: Invariance principle for the random conductance model in a degenerate ergodic environment,Ann. Probab.,43(2015), 1866–1891

work page 2015

[4] [4]

Armstrong, A

S. Armstrong, A. Bordas and J.-C. Mourrat: Quantitative stochastic homogenization and regularity theory of parabolic equations,Anal. PDE,11(2018), 1945–2014

work page 2018

[5] [5]

Superdiffusive central limit theorem for a Brownian particle in a critically-correlated incompressible random drift, September 2024

S. Armstrong, A. Bou-Rabee and T. Kuusi: Superdiffusive central limit theorem for a Brownian particle in a critically- correlated incompressible random drift, arXiv:2404.01115

work page arXiv

[6] [6]

Armstrong and P

S. Armstrong and P. Dario: Elliptic regularity and quantitative homogenization on percolation clusters,Comm. Pure Appl. Math.,71(2018), 1717–1849

work page 2018

[7] [7]

Armstrong and T

S. Armstrong and T. Kuusi: Renormalization group and elliptic homogenization in high contrast,Invent. Math.,242(2025), 895–1086. 22 XIN CHEN, CHENLIN GU AND JIAN W ANG

work page 2025

[8] [8]

Armstrong, T

S. Armstrong, T. Kussi and J.-C. Mourrat:Quantitative Stochastic Homogenization and Large-Scale Regularity, Grundlehren der mathematischen Wissenschaften, vol.352, Springer, Cham, 2019

work page 2019

[9] [9]

Armstrong and C.K

S. Armstrong and C.K. Smart: Quantitative stochastic homogenization of convex integral functionals,Ann. Sci. Éc. Norm. Supér.,48(2016), 423–481

work page 2016

[10] [10]

Armstrong and W

S. Armstrong and W. Wu:C 2 regularity of the surface tension for the∇ϕinterface model,Comm. Pure Appl. Math.,75 (2022), 349–421

work page 2022

[11] [11]

J. Bae, J. Kang, P. Kim and J. Lee: Heat kernel estimates for symmetric jump processes with mixed polynomial growths, Ann. Probab.,47(2019), 2830–2868

work page 2019

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Bercu, B

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work page 2015

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Berger and M

N. Berger and M. Biskup: Quenched invariance principle for simple random walk on percolation clusters,Probab. Theory Related Fields,137(2007), 83–120

work page 2007

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Biskup: Recent progress on the random conductance model,Prob

M. Biskup: Recent progress on the random conductance model,Prob. Surv.,8(2011), 294–373

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[15] [15]

Biskup, X

M. Biskup, X. Chen, T. Kumagai and J. Wang: Quenched invariance principle for a class of random conductance models with long-range jumps,Probab. Theory Related Fields,180(2021), 847–889

work page 2021

[16] [16]

Chatzigeorgiou, P

G. Chatzigeorgiou, P. Morfe, F. Otto and L. Wang: The Gaussian free-field as a stream function: asymptotics of effective diffusivity in infra-red cut-off,Ann. Probab.,53(2025), 1510–1536

work page 2025

[17] [17]

Chen, Z.-Q

X. Chen, Z.-Q. Chen, T. Kumagai and J. Wang: Homogenization of symmetric stable-like processes in stationary ergodic media,SIAM J. Math. Anal.,53(2021), 2957–3001

work page 2021

[18] [18]

Chen, Z.-Q

X. Chen, Z.-Q. Chen, T. Kumagai and J. Wang: Periodic homogenization of nonsymmetric Lévy-type processes,Ann. Probab., 49(2021), 2874–2921

work page 2021

[19] [19]

Chen, Z.-Q

X. Chen, Z.-Q. Chen, T. Kumagai and J. Wang: Quantitative stochastic homogenization for random conductance models with stable-like jumps,Probab. Theory Relat. Fields,191(2025), 627–669

work page 2025

[20] [20]

X. Chen, T. Kumagai and J. Wang: Random conductance models with stable-like jumps: quenched invariance principle,Ann. Appl. Probab.,31(2021), 1180–1231

work page 2021

[21] [21]

Z.-Q. Chen, P. Kim and T. Kumagai: Discrete approximation of symmetric jump processes on metric measure spaces,Probab. Theory Relat. Fields,155(2013), 703–749

work page 2013

[22] [22]

Dario and C

P. Dario and C. Gu: Quantitative homogenization of the parabolic and elliptic Green’s functions on percolation clusters,Ann. Probab.,49(2021), 556–636

work page 2021

[23] [23]

Fannjiang and T

A. Fannjiang and T. Komorowski: A martingale approach to homogenization of unbounded random flows,Ann. Probab.,25 (1997), 1872–1894

work page 1997

[24] [24]

Flegel, M

F. Flegel, M. Heida and M. Slowik: Homogenization theory for the random conductance model with degenerate ergodic weights and unbounded-range jumps,Ann. Inst. Henri Poincaré Probab. Stat.,55(2019), 1226–1257

work page 2019

[25] [25]

Funaki, C

T. Funaki, C. Gu and H. Wang: Quantitative homogenization and hydrodynamic limit of non-gradient exclusion process, arXiv:2404.12234, accepted byComm. Pure Appl. Math

work page arXiv

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Giunti, C

A. Giunti, C. Gu and J.-C. Mourrat: Quantitative homogenization of interacting particle systems,Ann. Probab.,50(2022), 1885–1946

work page 2022

[27] [27]

Gloria, S

A. Gloria, S. Neukamm and F. Otto: Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on Glauber dynamics,Invent. Math.,199(2015), 455–515

work page 2015

[28] [28]

Gloria and F

A. Gloria and F. Otto: An optimal variance estimate in stochastic homogenization of discrete elliptic equations,Ann. Probab., 39(2011), 779–856

work page 2011

[29] [29]

Gloria and F

A. Gloria and F. Otto: Quantitative results on the corrector equation in stochastic homogenization,J. Eur. Math. Soc.,19 (2017), 3489–3548

work page 2017

[30] [30]

Guo and O

X. Guo and O. Zeitouni: Quenched invariance principle for random walks in balanced random environment,Probab. Theory Related Fields,152(2012), 207–230

work page 2012

[31] [31]

Kassmann, A

M. Kassmann, A. Piatnitski and E. Zhizhina: Homogenization of Lévy-type operators with oscillating coefficients,SIAM J. Math. Anal.,51(2019), 3641–3665

work page 2019

[32] [32]

Kipnis and S

C. Kipnis and S. R.S. Varadhan: Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions,Comm. Math. Phys.,104(1986), 1–19

work page 1986

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Kozlov: The averaging of random operators,Mat

S.M. Kozlov: The averaging of random operators,Mat. Sb. (N.S.),109(1979), 188–202

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Kumagai:Random Walks on Disordered Media and Their Scaling Limits

T. Kumagai:Random Walks on Disordered Media and Their Scaling Limits. Lecture Notes in Mathematics/Ecole d’Ete de Probabilites de Saint-Flour, vol. 2101. Springer, Berlin, 2014

work page 2014

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Mathieu and A

P. Mathieu and A. Piatnitski: Quenched invariance principles for random walks on percolation clusters,Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.,463(2007), 2287–2307

work page 2007

[36] [36]

Papanicolaou and S.R.S

G.C. Papanicolaou and S.R.S. Varadhan: Boundary value problems with rapidly oscillating random coefficients. In:Random Fields, vol. I, II (Esztergom, 1979), volume 27 of Colloq. Math. Soc. János Bolyai, pp. 835–873. North-Holland, Amsterdam- New York, 1981

work page 1979

[37] [37]

Piatnitski and E

A. Piatnitski and E. Zhizhina: Stochastic homogenization of convolution type operators,J. Math. Pures Appl.,134(2020), 36–71

work page 2020

[38] [38]

Sidoravicius and A.-S

V. Sidoravicius and A.-S. Sznitman: Quenched invariance principles for walks on clusters of percolation or among random conductances,Probab. Theory Related Fields,129(2004), 219–244. QUANTITATIVE STOCHASTIC HOMOGENIZATION FOR RANDOM W ALKS WITH CRITICAL JUMP INDEX 23 Xin Chen:School of Mathematical Sciences, Shanghai Jiao Tong University, 200240 Shanghai,...

work page 2004