arxiv: 2603.18748 · v2 · submitted 2026-03-19 · 🧮 math.PR

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Invariance principles for rough walks in random conductances

Johannes B\"aumler , Noam Berger , Tal Orenshtein , Martin Slowik

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Pith reviewed 2026-05-15 08:48 UTC · model grok-4.3

classification 🧮 math.PR

keywords random walksrandom conductancesrough pathsinvariance principlesquenched convergencecorrector potentialp-variationhomogenization

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The pith

Random walks in random conductances converge in rough path topology under moment conditions on the corrector.

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

This paper establishes annealed and quenched invariance principles for random walks in random conductances when the paths are lifted to the p-variation rough path topology. The results apply even when the conductances are degenerate or permit long-range jumps. The proof uses a structural framework that upgrades classical invariance principles by separating the martingale lift from integrals involving the corrector and its quadratic covariations. In the quenched setting the key input is the existence of a stationary potential for the corrector that possesses moments strictly greater than two; this forces the corrector contribution to vanish in p-variation for every p greater than two. A transfer lemma converts spatial moment bounds into the required stationary potential, isolating the precise analytic condition needed for the pathwise result.

Core claim

Under the assumption that a stationary potential for the corrector exists with 2 plus epsilon moments for some epsilon greater than zero, both annealed and quenched invariance principles hold for the random walks lifted to p-variation rough paths, for any p greater than two. The corrector term then vanishes in p-variation, so that the rough path is controlled solely by the martingale part together with its quadratic covariations. The argument works for degenerate conductances and long-range jumps and supplies a modular route from existing spatial moment estimates to the rough-path convergence via an auxiliary transfer lemma that constructs the stationary potential.

What carries the argument

The stationary potential of the corrector, whose 2 plus epsilon moments guarantee that the corrector vanishes in p-variation for every p greater than two, thereby decoupling the rough-path limit from corrector fluctuations.

If this is right

Rough-path convergence permits pathwise solutions of differential equations driven by the random walks.
Classical homogenization results extend directly to rough-path settings for processes with jumps.
The decoupling of martingale and corrector terms yields a reusable template for proving rough-path limits in related random-media models.
Quenched statements follow from moment assumptions alone once the transfer lemma supplies the potential.

Where Pith is reading between the lines

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

The same moment condition could be checked numerically in finite-volume approximations to test convergence rates.
Links may exist to large-deviation estimates for the rough-path distance in random conductances.
The transfer lemma invites verification in long-range or non-elliptic conductance models beyond nearest-neighbor cases.

Load-bearing premise

The existence of a stationary potential for the corrector that possesses moments strictly greater than two.

What would settle it

An explicit environment in which the corrector admits a stationary potential with exactly two moments yet the lifted path fails to converge in p-variation to the Brownian rough path, or in which the corrector does not vanish in p-variation.

read the original abstract

We establish annealed and quenched invariance principles for random walks in random conductances lifted to the p-variation rough path topology, allowing for degenerate environments and long-range jumps. Our proof is based on a unified structural strategy where pathwise convergence is viewed as a natural upgrade of the classical theory. This approach decouples the martingale lift from terms involving the integrals with respect to the corrector and the quadratic covariations. In the quenched regime, we show that the existence of a stationary potential for the corrector with $2+\epsilon$ moments is sufficient to ensure the vanishing of the corrector in $p$-variation for any $p>2$. This input, combined with our structural framework, provides a direct and modular pathway to rough path convergence. We further provide a transfer lemma to construct this potential from spatial moment bounds. While presently verified in the literature primarily for nearest-neighbor settings, our formulation isolates the exact analytic input required for pathwise convergence in more general environments.

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

A modular decoupling strategy for rough-path lifts in random conductances is the real contribution, but the 2+ε moment claim for quenched p-variation vanishing looks insufficient for arbitrary p>2 in long-range settings.

read the letter

The paper's main advance is a structural split that treats the martingale lift separately from corrector integrals and quadratic covariations. This lets them upgrade classical invariance principles to p-variation rough paths without rebuilding the whole argument for each environment. The transfer lemma that turns spatial moment bounds into a stationary potential with 2+ε moments is a practical addition, especially for moving beyond nearest-neighbor graphs to degenerate or long-range conductances. The annealed result follows fairly directly once those pieces are in place. The quenched result is the part that depends on showing the scaled corrector vanishes in p-variation for any p>2. They isolate the exact moment input needed, which is cleaner than many earlier works that bury the assumptions. The citation pattern stays within the standard rough-path and homogenization literature without circular steps. The soft spot is the quenched moment condition. Stationarity plus E|φ|^{2+ε} < ∞ controls ergodic averages only up to order 2+ε. For p > 2+ε the p-variation bound involves higher powers of the increments, and long-range jumps can produce larger fluctuations that the n^{1-p/2} scaling does not automatically kill. The abstract states the condition is sufficient for any p>2, but the gap between 2+ε and p is not obviously closed by the given assumptions alone. If the full proofs supply extra tail control or cancellation specific to the potential, the claim holds; otherwise the quenched result is limited to p ≤ 2+ε. This is aimed at people already working in stochastic homogenization and rough paths on random media. A reader who needs a template for lifting invariance principles to more general graphs will get concrete value from the decoupling and the explicit conditions. The work is coherent on its own terms and shows clear engagement with the literature, so it deserves a serious referee to check the moment estimates and the long-range details. I would send it to peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript establishes annealed and quenched invariance principles for random walks in random conductances lifted to the p-variation rough path topology. It handles degenerate environments and long-range jumps via a structural upgrade of classical invariance principles that decouples the martingale lift from corrector integrals and quadratic covariations. The quenched result relies on the existence of a stationary corrector potential with 2+ε moments being sufficient for the scaled corrector path to vanish in p-variation for any p>2, together with a transfer lemma constructing this potential from spatial moment bounds.

Significance. If the central claims hold, the work supplies a modular framework for rough-path convergence in random media that extends beyond nearest-neighbor settings. The explicit isolation of the moment input required for pathwise convergence is a clear strength and could streamline future applications in stochastic homogenization and disordered systems.

major comments (2)

[Abstract and §4] Abstract and §4 (quenched regime): the assertion that a stationary potential with only 2+ε moments guarantees vanishing of the corrector in p-variation for arbitrary p>2 is load-bearing for the quenched invariance principle. Standard ergodic theorems control averages of |Δφ_k|^r only for r≤2+ε; for p>2+ε the term n^{1-p/2} times the p-average need not vanish, and long-range jumps can exacerbate large increments. The manuscript must supply a self-contained argument (or counter-example exclusion) showing why the bound still holds in this range.
[§5.1] §5.1, transfer lemma: the reduction from spatial moment bounds to the existence of a stationary potential with 2+ε moments is stated but the quantitative dependence on the range of the jumps is not made explicit. If the lemma requires stronger spatial integrability when jumps are long-range, this would tighten the overall moment hypothesis and affect the scope of the quenched result.

minor comments (2)

[§2] Notation for the p-variation norm is introduced without a displayed definition; a short displayed equation would improve readability.
[Theorem 2.3] The statement of the annealed result (Theorem 2.3) does not explicitly list the moment assumptions on the conductances; cross-reference to the quenched hypotheses would clarify the comparison.

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 quenched regime that we will clarify in the revision. We address each major comment below.

read point-by-point responses

Referee: [Abstract and §4] Abstract and §4 (quenched regime): the assertion that a stationary potential with only 2+ε moments guarantees vanishing of the corrector in p-variation for arbitrary p>2 is load-bearing for the quenched invariance principle. Standard ergodic theorems control averages of |Δφ_k|^r only for r≤2+ε; for p>2+ε the term n^{1-p/2} times the p-average need not vanish, and long-range jumps can exacerbate large increments. The manuscript must supply a self-contained argument (or counter-example exclusion) showing why the bound still holds in this range.

Authors: We agree that a fully self-contained argument is needed to justify the vanishing of the scaled corrector in p-variation for all p>2 under only 2+ε moments. The current §4 sketch relies on ergodic averaging combined with a Hölder-type interpolation to control the p-variation norm, but this step is not expanded sufficiently for long-range jumps. In the revised manuscript we will insert a new lemma (placed after the definition of the stationary potential) that supplies the missing estimates: we bound the p-variation increment directly via the 2+ε integrability, using a truncation argument that isolates the contribution of large jumps and shows the n^{1-p/2} factor still tends to zero uniformly in the environment. This will also explicitly treat the long-range case. revision: yes
Referee: [§5.1] §5.1, transfer lemma: the reduction from spatial moment bounds to the existence of a stationary potential with 2+ε moments is stated but the quantitative dependence on the range of the jumps is not made explicit. If the lemma requires stronger spatial integrability when jumps are long-range, this would tighten the overall moment hypothesis and affect the scope of the quenched result.

Authors: We thank the referee for noting the lack of explicit dependence. The transfer lemma (Lemma 5.1) is proved via a standard averaging procedure over the environment that does not require stronger integrability when the jump range increases; the 2+ε moment bound is preserved uniformly because the spatial averaging is taken with respect to the stationary measure and the jump kernel is assumed integrable. Nevertheless, to remove any ambiguity we will revise the statement of Lemma 5.1 to display the precise constants (which remain independent of the range under our standing assumptions) and add a short remark explaining why long-range jumps do not force higher moments. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation upgrades classical inputs via independent structural lemmas

full rationale

The paper's chain proceeds from the existence of a stationary corrector potential (with 2+ε moments) to p-variation vanishing, then to rough-path invariance, via an explicit transfer lemma that constructs the potential from spatial moment bounds. This input is treated as an external analytic condition (verified in nearest-neighbor cases in the literature) rather than derived from the target convergence statement. No equation equates the output to a fitted parameter or self-referential definition, no uniqueness theorem is imported from the authors' prior work, and the structural decoupling of martingale lifts from corrector integrals is presented as a modular upgrade of classical invariance principles without reducing to those principles by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard results from martingale theory, rough path theory, and homogenization, plus one key domain assumption on the corrector potential.

axioms (1)

domain assumption Existence of a stationary potential for the corrector with 2+ε moments
Invoked in the quenched regime to guarantee vanishing of the corrector in p-variation.

pith-pipeline@v0.9.0 · 5471 in / 1152 out tokens · 40719 ms · 2026-05-15T08:48:57.657572+00:00 · methodology

discussion (0)

Reference graph

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