pith. sign in

arxiv: 2604.21143 · v1 · submitted 2026-04-22 · 🧮 math.PR · math.AP

Quantitative homogenization for the critical long-range random conductance model

Ahmed Bou-Rabee , Paul Dario This is my paper

Pith reviewed 2026-05-09 23:00 UTC · model grok-4.3

classification 🧮 math.PR math.AP
keywords long-range random conductancecritical exponentquantitative homogenizationquenched invariance principleanomalous diffusionrandom walkBrownian motion
0
0 comments X

The pith

At the critical long-range jump exponent the random conductance model homogenizes to the Laplacian at rate 1/sqrt(|ln ε|) with diffusivity given by the mean conductance alone.

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

The paper studies the long-range random conductance model on the integer lattice where the jump rate between distant sites decays exactly as the critical power |x-y| to the minus (d+2). It proves that the associated discrete elliptic equation converges quantitatively to the ordinary Laplacian, with an error that decays as one over the square root of the absolute logarithm of the mesh size. From this it follows that the random walk converges almost surely to Brownian motion, but only after rescaling time by the slightly superdiffusive factor sqrt(t log t). The limiting diffusion constant equals the average conductance and receives no correction from any corrector field at leading order.

Core claim

For i.i.d. uniformly elliptic conductances a(x,y), the long-range operator with kernel a(x,y)|x-y|^{-(d+2)} homogenizes to a multiple of the discrete Laplacian at the quantitative rate 1/sqrt(|ln ε|). Consequently the quenched invariance principle holds under the scaling sqrt(t log t), and the effective diffusivity is exactly the mean of a with no leading-order corrector contribution.

What carries the argument

The critical long-range random conductance operator with |x-y|^{-(d+2)} decay together with the quantitative homogenization error estimate that controls the difference from the Laplacian without a leading corrector term.

If this is right

  • The random walk satisfies a quenched invariance principle under the scaling sqrt(t log t).
  • The effective diffusivity equals the spatial average of the conductances with no corrector correction.
  • Homogenization error is controlled at the explicit rate 1/sqrt(|ln ε|).
  • The model sits precisely between stable Lévy processes (faster decay) and standard Brownian motion (slower decay).

Where Pith is reading between the lines

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

  • The logarithmic divergence appears to suppress the usual corrector fluctuation that dominates error estimates in short-range homogenization.
  • The same scaling argument may extend to other lattice models whose jump kernels have exactly logarithmic second-moment divergence.
  • Direct Monte-Carlo simulation of the walk on grids of size 10^4 or larger could check whether the position variance grows as t log t.

Load-bearing premise

The conductances are independent and identically distributed with uniform positive lower and upper bounds, and the decay exponent is set exactly at d+2 so that the second moment of the jump kernel diverges only logarithmically.

What would settle it

Numerical solution of the elliptic equation on large finite boxes would falsify the claim if the observed homogenization error fails to decay proportionally to 1/sqrt(|ln ε|) or if the effective diffusivity deviates from the plain average conductance by an amount that does not vanish as the box size grows.

Figures

Figures reproduced from arXiv: 2604.21143 by Ahmed Bou-Rabee, Paul Dario.

Figure 1
Figure 1. Figure 1: A sample path of the critical long-range random walk on Z 2 with jump kernel |z| −4 and random conductances, shown at three successive scales: 500 steps (left), 5,000 steps (center), and 100,000 steps (right). By Theorem 1.2, the walk converges to Brownian motion under √ tlog t scaling. 1 arXiv:2604.21143v1 [math.PR] 22 Apr 2026 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Path counting (Lemma 2.3) for z = (3, 2) in Z 2 . The reference path γz→0 (blue) has |z|1 = 5 edges. Every path γv→u with v − u = z is a translate of γz→0. Two translates (green dashed, purple dotted) both use the target edge e (red), each because a different reference edge of γz→0 lands on e. Since each reference edge determines at most one translate, at most |z|1 paths use e. (2) For every η > 0 and ever… view at source ↗
Figure 3
Figure 3. Figure 3: The divergence-free cycle Cu+v,u for u = (0, 0) and v = (3, 2) in Z 2 . One unit of flow travels from u to u + v along the long-range edge (red, +1) and returns via the nearest-neighbour path γu+v→u (blue), which decreases coordinates lexicographically. The net flow at every vertex is zero. Fix a total order ≺ on εZ d (for instance, the lexicographic order), and define Iε :=  (u, v) ∈ U ε × (εZ d \ {0}) :… view at source ↗
Figure 4
Figure 4. Figure 4: The single-scale averaging argument (Lemma 4.2). The function h (red bump) is supported in U ε . Each application of Tk averages h at points shifted by ≈ ε3 k in the first coordinate. After N = ⌈diam(U)/(ε3 k )⌉ iterations, the sampled points lie outside U, so T N k h = 0 on U ε . The resulting prefactor N2 · (ε3 k ) 2 ε d = C diam(U) 2 ε d is independent of the scale k: this is special to the critical exp… view at source ↗
read the original abstract

We consider the long-range random conductance model on $\mathbb{Z}^d$ at the critical exponent: the jump rate between sites $x$ and $y$ decays as $\mathbf{a}(x,y) |x-y|^{-(d+2)}$, where $\mathbf{a}(x,y)$ are i.i.d. uniformly elliptic conductances. Below the critical exponent $(d+2)$ the walk converges to a stable process; above it, to Brownian motion with diffusive $\sqrt{t}$ scaling. At criticality the second moment of the jump kernel diverges logarithmically. We establish quantitative homogenization of the associated elliptic equation to the Laplacian at the rate $1/\sqrt{|\ln\varepsilon|}$. As a consequence, we deduce quenched convergence of the random walk to Brownian motion under the anomalous $\sqrt{t \log t}$ scaling. Unlike in standard homogenization, the effective diffusivity is determined by the mean conductance alone, with no corrector contribution at leading order.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript studies the long-range random conductance model on Z^d at the critical exponent, with i.i.d. uniformly elliptic conductances a(x,y) multiplying the kernel |x-y|^{-(d+2)}. It proves quantitative homogenization of the associated elliptic equation to the Laplacian at the explicit rate 1/sqrt(|ln ε|). As a consequence, it obtains quenched convergence of the random walk to Brownian motion under the anomalous scaling sqrt(t log t). The effective diffusivity equals the mean conductance with no leading-order corrector contribution, in contrast to standard homogenization.

Significance. If the central estimates hold, the result supplies a sharp quantitative bridge between subcritical stable Lévy processes and supercritical diffusive behavior, precisely capturing the logarithmic divergence of the second moment of the jump kernel. The explicit rate and the absence of a leading corrector are technically notable and may inform related models of anomalous diffusion in random media. The quenched invariance principle under the corrected scaling is a direct and falsifiable consequence once the homogenization error is controlled.

major comments (2)
  1. [§3] §3 (or the section containing the main homogenization theorem): the proof that the corrector contribution vanishes at leading order while still obtaining the 1/sqrt(|ln ε|) rate must be checked for circularity; the logarithmic divergence is controlled via the i.i.d. assumption, but it is unclear whether the variance estimates remain uniform when the second-moment integral is replaced by its truncated version.
  2. [Theorem 1.2] Theorem 1.2 (quenched convergence statement): the passage from the elliptic homogenization rate to the random-walk invariance principle under sqrt(t log t) scaling requires a quantitative control on the Green function or on the martingale approximation; the manuscript should exhibit the precise error term that converts the 1/sqrt(|ln ε|) bound into the claimed almost-sure convergence.
minor comments (2)
  1. [Abstract] The abstract and introduction should explicitly state whether the rate 1/sqrt(|ln ε|) is expected to be optimal (e.g., via a matching lower bound or heuristic) or only an upper bound.
  2. [§2] Notation for the rescaled conductances and the truncated kernels should be introduced once and used consistently; several places appear to reuse the symbol a_ε without redefinition.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments highlight important points on the logical structure of the estimates and the quantitative passage to the invariance principle. We address each major comment below and will revise the manuscript to incorporate clarifications and explicit error terms where needed.

read point-by-point responses

  1. Referee: [§3] §3 (or the section containing the main homogenization theorem): the proof that the corrector contribution vanishes at leading order while still obtaining the 1/sqrt(|ln ε|) rate must be checked for circularity; the logarithmic divergence is controlled via the i.i.d. assumption, but it is unclear whether the variance estimates remain uniform when the second-moment integral is replaced by its truncated version.

    Authors: The argument is not circular. The variance estimates for the corrector are first obtained for the truncated kernel using i.i.d. concentration inequalities (Bernstein-type bounds on the sum of independent terms), which hold uniformly in the truncation level because the truncation threshold is chosen after the logarithmic divergence is bounded by the mean conductance alone. The contribution of the tail is then controlled separately by the same 1/sqrt(|ln ε|) rate via a deterministic comparison to the averaged kernel. We will add a short paragraph at the beginning of §3 that explicitly states this order of estimates and the uniformity of the variance bound. revision: partial

  2. Referee: [Theorem 1.2] Theorem 1.2 (quenched convergence statement): the passage from the elliptic homogenization rate to the random-walk invariance principle under sqrt(t log t) scaling requires a quantitative control on the Green function or on the martingale approximation; the manuscript should exhibit the precise error term that converts the 1/sqrt(|ln ε|) bound into the claimed almost-sure convergence.

    Authors: We agree that the quantitative link deserves an explicit statement. In the proof of Theorem 1.2 the homogenization error enters the martingale approximation through the Green function representation of the position; the resulting error is of order 1/sqrt(log t) uniformly on compact time intervals after the sqrt(t log t) rescaling. Almost-sure convergence then follows from a Borel–Cantelli argument on dyadic times. We will insert a new lemma (or subsection) that records this error propagation and the precise constant dependence on the homogenization rate. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation proceeds from i.i.d. uniform ellipticity and the critical exponent (d+2) to control the homogenization error at rate 1/√|ln ε| via quantitative estimates that do not presuppose the target diffusivity or scaling. The claim that effective diffusivity equals the mean conductance (no leading corrector) is obtained as a direct consequence of the logarithmic divergence of the second moment together with the error bound, rather than by fitting or redefinition. The anomalous √(t log t) quenched convergence follows from the homogenization result by standard arguments. No load-bearing step reduces to a self-citation, ansatz smuggled via prior work, or renaming of a known pattern; the argument is self-contained against the stated assumptions and ergodicity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on i.i.d. uniform ellipticity of conductances and the mathematical fact of logarithmic second-moment divergence at exponent d+2; these are domain-standard assumptions with no new free parameters or invented entities.

axioms (2)
  • domain assumption Conductances a(x,y) are i.i.d. and uniformly elliptic, i.e., bounded between fixed positive constants λ and Λ independent of x,y.
    Explicitly stated in the abstract as the model setup.
  • standard math At the critical exponent d+2 the second moment of the jump kernel diverges logarithmically.
    This is a direct integral/sum computation for the kernel |x|^{-(d+2)} and is invoked to explain why standard diffusive scaling fails.

pith-pipeline@v0.9.0 · 5456 in / 1581 out tokens · 58133 ms · 2026-05-09T23:00:00.968742+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages

  1. [1]

    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

  2. [2]

    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, preprint, arXiv:2404.01115, 2024

    work page arXiv 2024

  3. [3]

    Armstrong, T

    S. Armstrong and T. Kuusi,Elliptic homogenization from qualitative to quantitative, preprint, arXiv:2210.06488, 2022

    work page arXiv 2022

  4. [4]

    Armstrong, T

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

    work page 2019

  5. [5]

    Armstrong and C

    S. Armstrong and C. Smart,Quantitative stochastic homogenization of convex integral functionals, Ann. Sci. ´Ec. Norm. Sup´ er.49(2016), 423–481. 28 AHMED BOU-RABEE AND PAUL DARIO

    work page 2016

  6. [6]

    R. F. Bass and D. A. Levin,Transition probabilities for symmetric jump processes, Trans. Amer. Math. Soc.354 (2002), 2933–2953

    work page 2002

  7. [7]

    Billingsley,Convergence of Probability Measures, 2nd ed., Wiley Series in Probability and Statistics, John Wiley & Sons, 1999

    P. Billingsley,Convergence of Probability Measures, 2nd ed., Wiley Series in Probability and Statistics, John Wiley & Sons, 1999

    work page 1999

  8. [8]

    Biskup,Recent progress on the random conductance model, Probab

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

    work page 2011

  9. [9]

    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 Fields180(2021), 847–889

    work page 2021

  10. [10]

    Bouchaud and A

    J.-P. Bouchaud and A. Georges,Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications, Phys. Rep.195(1990), 127–293

    work page 1990

  11. [11]

    Cannizzaro, Q

    G. Cannizzaro, Q. Moulard, and F. Toninelli,Superdiffusive central limit theorem for the stochastic Burgers equation at the critical dimension, preprint, arXiv:2501.00344, 2025

    work page arXiv 2025

  12. [12]

    Chen, Z.-Q

    X. Chen, Z.-Q. Chen, T. Kumagai, and J. Wang,Quenched invariance principle for long range random walks in balanced random environments, Ann. Inst. Henri Poincar´ e Probab. Stat.57(2021), 2243–2267

    work page 2021

  13. [13]

    Z.-Q. Chen, X. 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

  14. [14]

    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 Related Fields191(2025), 627–669

    work page 2025

  15. [15]

    Chen and T

    Z.-Q. Chen and T. Kumagai,Heat kernel estimates for jump processes of mixed types on metric measure spaces, Probab. Theory Related Fields140(2008), 277–317

    work page 2008

  16. [16]

    Z.-Q. Chen, T. Kumagai, and J. Wang,Heat kernel estimates for general symmetric pure jump Dirichlet forms, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)23(2022), 1091–1140

    work page 2022

  17. [17]

    X. Chen, T. Kumagai, and J. Wang,Random conductance models with stable-like jumps: heat kernel estimates and Harnack inequalities, J. Funct. Anal.279(2020), 108656

    work page 2020

  18. [18]

    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

  19. [19]

    S. N. Ethier and T. G. Kurtz,Markov Processes: Characterization and Convergence, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, 1986

    work page 1986

  20. [20]

    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 Poincar´ e Probab. Stat.55(2019), 1226–1257

    work page 2019

  21. [21]

    Gilbarg and N

    D. Gilbarg and N. S. Trudinger,Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer, 2001

    work page 2001

  22. [22]

    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

  23. [23]

    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

  24. [24]

    Gloria and F

    A. Gloria and F. Otto,An optimal error estimate in stochastic homogenization of discrete elliptic equations, Ann. Appl. Probab.22(2012), 1–28

    work page 2012

  25. [25]

    Kassmann, A

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

    work page 2019

  26. [26]

    S. M. Kozlov,Averaging of random operators, Mat. Sb. (N.S.)109(151)(1979), no. 2, 188–202

    work page 1979

  27. [27]

    Murugan and L

    M. Murugan and L. Saloff-Coste,Anomalous threshold behavior of long range random walks, Electron. J. Probab. 20(2015), no. 74, 1–21

    work page 2015

  28. [28]

    Murugan and L

    M. Murugan and L. Saloff-Coste,Davies’ method for anomalous diffusions, Proc. Amer. Math. Soc.145(2017), 1793–1804

    work page 2017

  29. [29]

    Murugan and L

    M. Murugan and L. Saloff-Coste,Heat kernel estimates for anomalous heavy-tailed random walks, Ann. Inst. Henri Poincar´ e Probab. Stat.55(2019), 697–719

    work page 2019

  30. [30]

    G. C. Papanicolaou and S. R. S. Varadhan,Boundary value problems with rapidly oscillating random coefficients, in: Random Fields, Vol. I, II (Esztergom, 1979), Colloq. Math. Soc. J´ anos Bolyai, vol. 27, North-Holland, Amsterdam, 1981, pp. 835–873

    work page 1979

  31. [31]

    Piatnitski and E

    A. Piatnitski and E. Zhizhina,Periodic homogenization of nonlocal operators with a convolution-type kernel, SIAM J. Math. Anal.49(2017), 64–81

    work page 2017

  32. [32]

    R. W. Schwab,Periodic homogenization for nonlinear integro-differential equations, SIAM J. Math. Anal.42 (2010), 2652–2680

    work page 2010

  33. [33]

    Sz´ asz and T

    D. Sz´ asz and T. Varj´ u,Limit laws and recurrence for the planar Lorentz process with infinite horizon, J. Stat. Phys.129(2007), 59–80

    work page 2007

  34. [34]

    V. V. Yurinski˘ ı,On a Dirichlet problem with random coefficients, in: Stochastic Differential Systems (Proc. IFIP-WG 7/1 Working Conf., Vilnius, 1978), Lecture Notes in Control and Information Sci., vol. 25, Springer, Berlin–New York, 1980, pp. 344–353. Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104 Email address:ahmedmb@sa...

    work page 1978