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Anchored Nash inequalities and heat kernel bounds for a class of random conductance models with long-range jumps

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Pith's one-line read Degenerate long-range random walks still decay like t^{-d/2} on good graphs.

desk verdict A solid extension of Mourrat–Otto's anchored Nash method to non-local, degenerate, long-range random conductance models, with honest limitations and a clean proof structure. read the letter →

arxiv 2607.14718 v1 pith:XKXCDG3T submitted 2026-07-16 math.PR math.AP

Sebastian Andres , Xin Chen , Martin Slowik , Kun Yin This is my paper · ORCID
classification math.PRmath.AP MSC 60K3760F1782C4182B43
keywords Nashinequalityrandomconductancemodelheatkernellong-rangejumpspercolationclustersdegenerateweightsDirichletformon-diagonalbound
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

Anchored Nash inequalities are the paper's engine: they control the L^2 norm of a function by a non-local Dirichlet form, the L^1 norm, and a weighted L^2 norm with a polynomial weight. The paper proves them for divergence-form operators with degenerate, possibly unbounded jump rates and long-range jumps, on any graph with large-scale volume regularity and a Poincaré-Sobolev inequality. It then converts these inequalities into on-diagonal heat kernel upper bounds of the form p(t,o,o) ≤ C t^{-d/2} for d≥3, both quenched and annealed. This covers supercritical percolation clusters, including correlated models, giving the first global functional inequality of this kind on such degenerate graphs. The results extend earlier nearest-neighbour anchored Nash theory to non-local, non-uniformly elliptic settings with general speed measures.

What carries the argument

The interpolated anchored Nash inequality (1.13), ∥f∥²₂ ≤ C M E(f)^α ∥f∥^{2β}_1 ∥η^{m/2}f∥^γ_2, with α+β+γ=1, is the central object. It controls the L² norm without any uniform ellipticity; the cost is an extra weighted norm with the polynomial weight η. Carrying the proof are the large-scale Poincaré-Sobolev inequality (1.7), a time-change to a process with speed measure π_{m0}=max{1, μ_{m0}, θ} that satisfies the two-sided jump controls (3.1)–(3.2), and an optimisation lemma (Lemma 2.2) that turns the three-term estimate into the Nash form.

What would settle it

Find a graph satisfying the volume growth (1.5) but violating the Poincaré–Sobolev inequality (1.7), and exhibit a degenerate long-range walk on it whose on-diagonal heat kernel decays slower than t^{-d/2}; this would show the anchored Nash inequality cannot be necessary. Alternatively, exhibit a correlated supercritical percolation model for which the relative isoperimetric inequality (1.32) fails; Theorems 1.14–1.15 would then predict a heat kernel bound that cannot hold.

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Extended reading notes

Core claim

The central claim is that a degenerate random walk with long-range jumps on a graph satisfying Assumption 1.1 has a diffusive on-diagonal heat kernel: for d≥3 and t>0, p(t,o,o) ≤ C t^{-d/2}, where C depends on the origin, the graph constants, and the integrability of the speed measure, the jump measure, and the inverse jump rates. The proof builds an interpolated anchored Nash inequality, which remains valid even when the classical Nash or Sobolev inequality fails, and then transfers the heat kernel bound to the original walk by a time-change argument.

Load-bearing premise

The paper's conclusions stand or fall on the large-scale Poincaré–Sobolev inequality (1.7) on every ball; for percolation clusters, this is imported as the relative isoperimetric inequality (1.32), whose verification is cited from the literature rather than proved here.

Editorial extensions

If this is right

  • Quenched on-diagonal heat kernel bounds of order t^{-d/2} hold for long-range random conductance models on Z^d under explicit moment conditions on θ, μ_{m0}, and ν.
  • Annealed heat kernel bounds of the same order hold for d≥3, and with a logarithmic correction for d=2, under strengthened moment conditions.
  • The bounds apply to supercritical percolation clusters, including correlated percolation models, giving the first global Nash-type inequality on such degenerate graphs.
  • The random constants in the quenched bound are explicit, which is what makes the annealed bounds follow from the maximal ergodic theorem.
  • For nearest-neighbour walks the bound holds for any m0>d, removing the restriction p ≥ d/(m0−2) needed for long-range jumps.

Reading between the lines

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

  • If the anchored Nash inequality is robust, it should also yield off-diagonal or near-diagonal bounds by Davies-type perturbations, although the authors note the current constant depends on the origin and prevents near-diagonal bounds in this form.
  • The annealing step suggests a template: any model where the environment seen from the particle is ergodic and the moment conditions (1.27)–(1.28) hold should inherit t^{-d/2} annealed decay in d=2 with the same logarithmic correction.
  • The percolation application rests entirely on the relative isoperimetric inequality (1.32); verifying that inequality for new correlated percolation models (e.g., level sets of random fields) would immediately extend Theorems 1.14–1.15 to those models.
  • One could test the optimality of the moment conditions: if the bound fails when 1/p+1/q = 2/d, the integrability threshold is sharp.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper proves anchored Nash inequalities for discrete non-local divergence-form operators with degenerate weights and uses them to derive on-diagonal heat-kernel upper bounds for random conductance models with long-range jumps. Theorem 1.3 establishes an anchored Nash inequality under an integrability condition on the reference measure θ, its reciprocal, and the dual weight ν, on graphs satisfying a large-scale volume-regularity and Poincaré–Sobolev assumption (Assumption 1.1). Theorem 1.4 upgrades this to an interpolated three-term inequality, and Theorem 1.5 transfers the inequality to a heat-kernel bound of the form p(t,o,o) ≤ C t^{-d/2} for d ≥ 3, via a time change to the speed measure π_{m0} = max{1, μ_{m0}, θ}. Applications include quenched and annealed bounds on Z^d in ergodic environments (Theorems 1.8 and 1.9) and quenched bounds on supercritical percolation clusters, including correlated models (Theorems 1.14 and 1.15), conditional on a scale-uniform relative isoperimetric inequality imported from the literature.

Significance. If the results are correct, the paper gives a substantial extension of the Mourrat–Otto anchored-Nash program: it removes uniform ellipticity for non-local operators, allows a general reference measure, and produces explicit constants that make annealed estimates available. The proofs are detailed and the algebraic identities in Lemma 2.2 and Proposition 3.1 check out. A particular strength is the time-change argument, which makes the auxiliary conditions (3.1)–(3.2) automatic for π_{m0}, and the explicit dependence of the constants on M0, R0 and R1, which is what enables Theorem 1.9. The percolation applications are more conditional: they rely on an isoperimetric input (Assumption 1.12(iii)) that is cited rather than proved in this paper.

major comments (2)
  1. [§3.2, proof of Theorem 1.5(i)] The invocation of Lemma 3.3 is not literally consistent as written. In Lemma 3.3 the exponent q is the one controlling ∥μ_{m0}∥ in (3.10), while in Theorem 1.5(i) the assumption (1.14) controls μ_{m0} in L^p and only ν in L^q. The sentence defining K2 with sup_R ∥μ_{m0}∥_{p,B(R)} and exponent R0^{1/p+1/q} therefore does not match the lemma's hypotheses unless one silently renames the exponents: take the lemma's q equal to the theorem's p and the lemma's p equal to ∞ (which is allowed because π_{m0}^{-1} ≤ 1). The final bound is correct, but the proof should state this exponent choice explicitly; otherwise the reader cannot verify the application of Lemma 3.3.
  2. [§1.4 and Theorem 1.15] The percolation results are conditional on Assumption 1.12(iii), the scale-uniform relative isoperimetric inequality (1.32). For Bernoulli percolation this is cited to [11,39], and for correlated models to [43]; it is not proved here. Since the abstract claims results on 'possibly correlated supercritical percolation clusters', the paper should state plainly in Theorem 1.15 or Remark 1.13 that the validity for a given correlated model is exactly the validity of (1.32), and that no general verification is contained in this manuscript. This is not an internal error, but the scope statement should be tightened.
minor comments (5)
  1. [Remark 1.6(v)] The phrase 'near-diagonal bounds of the form p(t,o,x) ≲ t^{-d/2}, x∈V' is off-diagonal rather than near-diagonal; the terminology should be adjusted.
  2. [§3.2, proof of Theorem 1.5(i)] In the same passage, the notation reuses p both for the theorem's exponent and implicitly for the lemma's q. Rewriting with p_μ, q_ν or a similar convention would remove the ambiguity.
  3. [§5, proof of Theorems 1.14–1.15] The final step says Assumption 1.1 follows from Proposition 5.1 and Assumption 1.12. This is plausible, but the passage from the Q^ω(x,r)-boxes in (5.1) to the graph balls B^ω(x,r) in (1.7) is compressed; a sentence explaining the covering argument would help.
  4. [Lemma 3.3] The constants in the proof of Lemma 3.3 use the volume bounds for B(R) with R≥R0; the statement assumes (1.5) but not explicitly that R0 is the constant from Assumption 1.1. This is clear from context, but the sentence could be made precise.
  5. [General] There are minor typographical issues, e.g. 'HEA T KERNEL' in the title and 'comparatable' in Remark 1.13; these should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: main bound is derived from explicit assumptions with full proofs; external percolation inputs are cited, not self-referential.

full rationale

The derivation chain is self-contained from Assumption 1.1. Theorem 1.3 is proved from (1.7), volume growth, and integrability (1.10) without assuming the conclusion; Theorem 1.4 follows by an explicit covering argument and Lemma 2.2. Theorem 1.5 is obtained by checking (3.1)-(3.2) for the time-changed process with pi_m0 (3.12), applying Proposition 3.1, and transferring the bound via (3.16)-(3.17); no fitted parameter is renamed as a prediction and no inequality reduces to an input by construction. The self-citations ([18], [22], [4], [5]) are contextual or comparative, and the main heat-kernel argument follows [40] with details supplied. The only genuinely external load-bearing input is Assumption 1.12(iii), the isoperimetric inequality (1.32) for correlated percolation, imported from [43]/[11]/[39]; this is an independent geometric input, not a restatement of the target heat-kernel bound, so it is a correctness risk rather than circularity.

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

The central claims rest on a large-scale Poincaré–Sobolev inequality (Assumption 1.1), explicit integrability/moment conditions, and, for percolation, geometric Assumption 1.12 imported from [11,39,43]. No parameter fitting and no invented entities; the contribution is a derivation of heat kernel bounds from these stated assumptions.

assumptions (5)
  • domain assumption Assumption 1.1: large-scale volume regularity (1.5) and Poincaré–Sobolev inequality (1.7) on all balls B(x,r).
    Used throughout Section 2 to prove the anchored Nash inequality; must hold for the graph (verified for Z^d, imported for percolation clusters).
  • domain assumption Integrability conditions (1.10)/(1.14): sup over balls of normalized L^p norms of θ, θ^{-1}, ν, μ_{m0}.
    Controls degeneracy of weights and jump kernel; failing this, trapping can break Gaussian decay (cf. [16]).
  • domain assumption Assumption 1.7: stationarity and ergodicity of the environment and of θ for the RCM applications.
    Needed for quenched/annealed bounds and for application of the maximal ergodic theorem in Section 4.
  • domain assumption Assumptions 1.11 and 1.12 for percolation: unique infinite cluster, volume regularity, distance comparability, isoperimetric inequality (1.32).
    For percolation applications; cited from [11], [39], [43] and not proved in this paper.
  • standard math Standard background: Carlen–Kusuoka–Stroock equivalence of Nash inequalities and heat kernel bounds, Dirichlet form theory, maximal ergodic theorem.
    Invoked without proof as standard results.

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Pith. "Pith review of Anchored Nash inequalities and heat kernel bounds for a class of random conductance models with long-range jumps." pith.science (2026). https://pith.science/paper/XKXCDG3T

@misc{pith2026260714718,
  author       = {Pith},
  title        = {Pith review of: Anchored Nash inequalities and heat kernel bounds for a class of random conductance models with long-range jumps},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XKXCDG3T}},
  note         = {Machine review of arXiv:2607.14718}
}
abstract

We show anchored versions of the Nash inequality for discrete non-local divergence-form operators with degenerate weights. They allow to control the $L^{2}$-norm of a function by Dirichlet forms that are not uniformly elliptic. We then use them to provide on-diagonal heat kernel upper bounds for a class of random conductance models with degenerate jump rates allowing long-range jumps. The results are established on a class of graphs including the integer lattice and possibly correlated supercritical percolation clusters.

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