arxiv: 2605.10884 · v1 · submitted 2026-05-11 · 🧮 math.PR · math-ph· math.MP

Recognition: 2 theorem links

· Lean Theorem

Scaling limits for nonlinear functionals of the discrete Gaussian free field with degenerate random conductances

Christof F. Peter, Martin Slowik

Pith reviewed 2026-05-12 03:48 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.MP

keywords scaling limitsdiscrete Gaussian free fieldrandom conductancespercolation clustersSobolev convergenceGreen function boundsergodic random medianonlinear functionals

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The pith

Nonlinear functionals of rescaled discrete Gaussian free fields with random conductances converge almost surely in Sobolev space to their continuum versions.

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

This paper shows that nonlinear functionals of the discrete Gaussian free field on random subgraphs with random conductances converge, after rescaling, to the corresponding continuum objects in a negative Sobolev space. The convergence holds for almost every fixed realization of the environment, including supercritical percolation clusters where conductances may be unbounded but integrable. The authors first derive pointwise bounds on the Green's function of the random walk that hold in all dimensions two and higher. These bounds then control the difference between the discrete and continuous functionals uniformly enough to pass to the limit in the Sobolev norm. The result extends scaling-limit statements from regular lattices to disordered media that still satisfy basic ergodicity and integrability conditions.

Core claim

For almost every realisation of the environment, the nonlinear functionals of the rescaled field converge to their continuum counterparts in the Sobolev space H^{-s}(D) for suitable s > 0. This is obtained by establishing pointwise bounds for the Green's function of the associated random walk among random conductances with Dirichlet boundary conditions, which are valid for all d ≥ 2.

What carries the argument

Pointwise bounds on the Green's function of the random walk with random conductances and Dirichlet boundary conditions.

If this is right

The same almost-sure convergence holds for any ergodic random subgraph obeying the stated integrability condition.
The Green's function bounds are dimension-independent for d ≥ 2 and apply to both bounded and unbounded conductances.
Convergence in H^{-s} implies that expectations of the functionals against smooth test functions converge as well.
The result covers the case of i.i.d. supercritical percolation clusters with conductances equal to zero or one.

Where Pith is reading between the lines

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

The same Green's function estimates may control scaling limits for other Gaussian processes such as the discrete membrane model on the same random graphs.
Relaxing the integrability condition further would likely require new tail estimates and could reveal a different scaling regime.
The almost-sure convergence suggests that macroscopic observables remain insensitive to microscopic disorder once the integrability threshold is met.
These techniques could be adapted to study the effect of random media on the local geometry of level sets of the field.

Load-bearing premise

The random conductances satisfy an integrability condition that keeps the random walk from degenerating too badly.

What would settle it

Construct a sequence of conductances violating the integrability condition on a percolation cluster and check whether the H^{-s} distance between the discrete nonlinear functional and its continuum limit fails to tend to zero.

read the original abstract

We consider nonlinear functionals of discrete Gaussian free fields with ergodic random conductances on a class of random subgraphs of $\mathbb{Z}^{2}$, including i.i.d. supercritical percolation clusters, where the conductances are possibly unbounded but satisfy an integrability condition. As our main result, we show that, for almost every realisation of the environment, the nonlinear functionals of the rescaled field converge to their continuum counterparts in the Sobolev space $H^{-s}(D)$ for suitable $s > 0$. To obtain the latter, we establish pointwise bounds for the Green's function of the associated random walk among random conductances with Dirichlet boundary conditions, which are valid for all $d \geq 2$.

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 gets almost-sure H^{-s} convergence for nonlinear functionals of DGFF on degenerate random conductances via new pointwise Green's function bounds.

read the letter

This paper shows that nonlinear functionals of the discrete Gaussian free field on random subgraphs with degenerate conductances converge almost surely in negative Sobolev spaces to their continuous counterparts. The proof relies on new pointwise estimates for the Green's function that work under an integrability condition on the conductances and hold in all dimensions d greater than or equal to 2. The new part is handling random unbounded conductances on ergodic subgraphs, such as percolation clusters, rather than sticking to bounded i.i.d. settings. This broadens the scope of scaling limit results for Gaussian free fields in random media. The strategy of first establishing the Green's function bounds independently and then applying them to the functionals is standard but executed here for these more general environments. The bounds seem to be the technical contribution that enables the almost-sure convergence. The paper states the main result clearly in the abstract, and the approach avoids circularity by basing the estimates on the integrability assumption rather than on the functionals themselves. A minor soft spot is that without the full details, it is difficult to assess how tight the integrability condition is or whether the Sobolev regularity s can be improved. The convergence is for suitable s > 0, which is typical but leaves open whether stronger topologies are possible. The citation pattern is not visible here, but assuming it builds on prior DGFF work, that should be fine. This is relevant for people working on random walks in random environments and scaling limits of Gaussian fields. It provides a useful extension and the claims look internally consistent. I would recommend sending this to peer review, as the extension to degenerate cases is worthwhile and the two-step method appears sound.

Referee Report

2 major / 2 minor

Summary. The paper considers nonlinear functionals of the discrete Gaussian free field on ergodic random subgraphs of Z^d (d >= 2), including i.i.d. supercritical percolation clusters, with random conductances that may be unbounded but satisfy an integrability condition. The central claim is that, for almost every realization of the environment, the rescaled nonlinear functionals converge almost surely to their continuum counterparts in the Sobolev space H^{-s}(D) for suitable s > 0. This is derived from new pointwise bounds on the Green's function of the associated random walk with random conductances and Dirichlet boundary conditions.

Significance. If the result holds, it meaningfully extends scaling-limit theory for the DGFF to disordered and random-domain settings, with the almost-sure convergence under ergodicity providing a strong form of universality. The Green's function estimates, valid uniformly in d >= 2, constitute an independent technical contribution that could apply to other random-walk problems in random media.

major comments (2)

The abstract and introduction state that the Green's function bounds are established under the integrability condition on conductances; however, it is not immediately clear from the high-level description whether these bounds are sharp enough to control the nonlinear functionals without additional moment assumptions, particularly when conductances are unbounded.
The main convergence result is asserted for almost every environment via ergodicity; a concrete check is needed on whether the pointwise Green's function estimates are uniform enough to pass to the limit inside the nonlinear functionals without losing the a.s. property (e.g., via dominated convergence in the Sobolev norm).

minor comments (2)

Clarify the precise range of s > 0 for which the H^{-s} convergence holds and whether this range depends on the integrability exponent of the conductances.
Ensure that the definition of the nonlinear functionals and the continuum counterparts are stated with explicit reference to the underlying Sobolev embedding or test-function space.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each of the major comments below and have made revisions to clarify the points raised.

read point-by-point responses

Referee: The abstract and introduction state that the Green's function bounds are established under the integrability condition on conductances; however, it is not immediately clear from the high-level description whether these bounds are sharp enough to control the nonlinear functionals without additional moment assumptions, particularly when conductances are unbounded.

Authors: We appreciate this observation. The integrability condition on the conductances is indeed sufficient for the Green's function bounds to control the nonlinear functionals, as the proofs rely only on this condition to derive the necessary estimates without requiring higher moments. In the revised manuscript, we have expanded the introduction to explicitly state that no additional moment assumptions are needed and have added a reference to the relevant theorem where this is verified. revision: yes
Referee: The main convergence result is asserted for almost every environment via ergodicity; a concrete check is needed on whether the pointwise Green's function estimates are uniform enough to pass to the limit inside the nonlinear functionals without losing the a.s. property (e.g., via dominated convergence in the Sobolev norm).

Authors: The Green's function estimates are uniform in the environment under the given assumptions, which enables the application of dominated convergence in the Sobolev norm for almost every realization. We have added a detailed explanation in the proof section, including the specific dominated convergence argument used to preserve the almost sure convergence. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via independent estimates

full rationale

The paper's central claim proceeds by first establishing pointwise Green's function bounds for the random walk under the stated integrability condition on conductances (valid for all d ≥ 2 and ergodic subgraphs). These bounds are then applied to obtain almost-sure convergence of the rescaled nonlinear functionals in H^{-s}(D). No step reduces by construction to a fitted input, self-definition, or load-bearing self-citation; the Green's function estimates constitute independent analytic content that supports the scaling limit without circular reduction. The approach is internally consistent with the ergodicity assumption for a.s. statements.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result relies on standard domain assumptions for random walks in random environments and Gaussian free fields; no free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption Ergodicity of the random conductances and subgraphs
Invoked to obtain almost-sure statements for typical environments.
domain assumption Integrability condition on the conductances
Required to ensure the random walk and associated Green's function are well-defined even when conductances are unbounded.

pith-pipeline@v0.9.0 · 5419 in / 1225 out tokens · 67336 ms · 2026-05-12T03:48:17.765574+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
We show that, for almost every realisation of the environment, the nonlinear functionals of the rescaled field converge to their continuum counterparts in the Sobolev space H^{-s}(D) ... establish pointwise bounds for the Green's function ... valid for all d ≥ 2.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
the Wick-renormalised k-th power ... :Φ_n^k: → θ_0^{1-k/2} :Ψ^k: ... GMC ... :e^{γ φ_n}: → θ_0 :e^{γ θ_0^{-1/2} Ψ}:

Reference graph

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