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arxiv: 2603.13798 · v2 · pith:EZKEITPEnew · submitted 2026-03-14 · 🧮 math.PR · math-ph· math.CO· math.MP

Iterated Graph Systems (I): random walks and diffusion limits

Ziyu Neroli This is my paper

Pith reviewed 2026-05-21 10:47 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.COmath.MP
keywords random walksdiffusion limitsfractal graphsEdge Iterated Graph Systemsheat kernel estimatesGromov-Hausdorff topologypercolation cluster
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The pith

Rescaled simple random walks on Edge Iterated Graph Systems converge to limiting diffusions in the Gromov-Hausdorff-Prokhorov-Skorokhod topology.

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

This paper shows that for graphs built from Edge Iterated Graph Systems, rescaled simple random walks converge in a refined topology to a diffusion limit. The limit matches Brownian motion exactly when the resistance dimension is positive. By identifying the degree dimension as the right correction for heat-kernel estimates, the work unifies the treatment of locally finite graphs and locally infinite scale-free graphs. This matters because it provides a single framework for diffusion on fractals and settles a long-standing question about percolation on the diamond hierarchical lattice.

Core claim

The paper proves that the rescaled simple random walks converge in the Gromov--Hausdorff--Prokhorov--Skorokhod topology to the limiting diffusion. This diffusion coincides with Brownian motion when the resistance dimension is positive. The graph analysis shows that the degree dimension serves as the natural correction term for on-diagonal heat-kernel estimates, yielding a unified formulation that works in both the locally finite and the locally infinite regimes.

What carries the argument

Edge Iterated Graph Systems (EIGS) generating fractal graphs, using the degree dimension to correct heat-kernel estimates and the Gromov-Hausdorff-Prokhorov-Skorokhod topology to establish convergence.

If this is right

  • The result gives convergence in a topology that combines metric, measure and path properties.
  • The diffusion limit is Brownian motion when the resistance dimension is positive.
  • The same heat-kernel correction applies to both locally finite and locally infinite graphs.
  • It provides the solution to the open problem on the DHL percolation cluster.

Where Pith is reading between the lines

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

  • The approach may generalize to other classes of self-similar graphs not generated by EIGS.
  • Similar dimension corrections could apply to higher-order processes like Levy flights on fractals.
  • This could lead to new numerical methods for simulating diffusions on scale-free networks.

Load-bearing premise

The structural properties of the Edge Iterated Graph Systems allow the degree dimension to serve as the natural correction term for on-diagonal heat-kernel estimates in both locally finite and locally infinite regimes.

What would settle it

Finding an explicit Edge Iterated Graph System where the rescaled simple random walks fail to converge in the Gromov-Hausdorff-Prokhorov-Skorokhod topology to the predicted diffusion would falsify the result.

Figures

Figures reproduced from arXiv: 2603.13798 by Ziyu Neroli.

Figure 1
Figure 1. Figure 1: Example of EIGS: diamond hierarchical lattice (DHL) [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: An example: Canonical Xi graph (in Z 2 ) Definition 1.1. See [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: An example of two-coloured EIGS specified as follows. Let K ∈ N be the number of edge colours. For every graph G that appears in the construction, let C : E(G) −→ [K] be a colour map. 1. Initial graph. Ξ 0 is an edge-coloured finite directed graph. If Ξ 0 is just an edge, we denote its two vertices by v+ and v−. In particular, Ξ 0 ι denotes the special case that the initial graph is just one ι-coloured dir… view at source ↗
Figure 4
Figure 4. Figure 4: (u, v)-flower Example 2.29. Fix integers u ≥ 2 and v ≥ 2. The (u, v)-flower is the single-coloured EIGS obtained by replacing every edge by u pairwise edge-disjoint parallel paths, each of graph length v, with the planting vertices identified with the two endpoints of the original edge. See [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: A sample trace of a 106 -step simple random walk on the level-6 approximation Ξ 6 of the canonical Xi graph, drawn in its natural embedding in Z 2 . The underlying graph is shown in grey and the trace in red [PITH_FULL_IMAGE:figures/full_fig_p028_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: (2, 2)-flower [PITH_FULL_IMAGE:figures/full_fig_p033_7.png] view at source ↗
Figure 10
Figure 10. Figure 10: The random EIGS associated with percolation [PITH_FULL_IMAGE:figures/full_fig_p033_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: A critical percolation cluster (red subgraph) on DHL generated by random EIGS. [PITH_FULL_IMAGE:figures/full_fig_p034_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Reduction of the two-terminal resistance problem to a random EIGS. [PITH_FULL_IMAGE:figures/full_fig_p035_12.png] view at source ↗
read the original abstract

This paper investigates random walks and diffusion limits on a broad class of fractal graphs generated by Edge Iterated Graph Systems (EIGS). We prove that the rescaled simple random walks converge in the Gromov--Hausdorff--Prokhorov--Skorokhod topology to the limiting diffusion, which coincides with Brownian motion when the resistance dimension is positive. The graph analysis underlying this convergence identifies the degree dimension as the natural correction term for on-diagonal heat-kernel estimates, yielding a unified formulation in the locally finite and locally infinite (scale-free) regimes. Using this framework, we solve the open problem on the DHL percolation cluster posed by Hambly and Kumagai [Commun. Math. Phys. 295 (2010), 29--69].

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 paper introduces Edge Iterated Graph Systems (EIGS) as a framework for generating fractal graphs and proves that rescaled simple random walks on these graphs converge in the Gromov-Hausdorff-Prokhorov-Skorokhod topology to a limiting diffusion process. This limit coincides with Brownian motion when the resistance dimension is positive. The analysis identifies the degree dimension as the correction term for on-diagonal heat-kernel estimates, providing a unified treatment across locally finite and locally infinite (scale-free) regimes, and applies the framework to resolve the open problem on the DHL percolation cluster posed by Hambly and Kumagai (Commun. Math. Phys. 2010).

Significance. If the central convergence result holds, the work offers a meaningful extension of diffusion-limit theorems to graphs with unbounded degrees, solving a concrete open problem in the literature on random walks and analysis on fractals. The introduction of EIGS as a generative model and the explicit use of degree dimension for heat-kernel corrections are strengths that could support further applications in percolation and scale-free network analysis.

major comments (2)
  1. [§4, Theorem 4.3] §4, Theorem 4.3 and the heat-kernel estimates in §3.2: the on-diagonal bound p_{2n}(x,x) ∼ n^{-d_deg/d_w} is asserted to hold uniformly in both locally finite and locally infinite EIGS regimes, but the argument for the scale-free case (unbounded degrees) does not appear to supply a uniform control on maximal degree or logarithmic growth that would guarantee the required volume-doubling and Hölder constants. Without this, an extra slowly-varying factor can enter the estimate and propagate into the tightness and identification steps for the GHPS-Skorokhod limit, undermining the claim that the limit is the resistance diffusion.
  2. [§6] §6 (application to DHL percolation cluster): the resolution of the Hambly-Kumagai open problem is presented as a corollary of the general EIGS theory, yet the manuscript does not explicitly verify that the percolation cluster satisfies the structural hypotheses of an EIGS (in particular the edge-iteration rules and the resulting degree dimension) or isolate which precise statement from the 2010 paper is now settled.
minor comments (2)
  1. [Abstract and §1] The abstract and §1 would benefit from a one-sentence clarification of which specific open question (e.g., existence of the diffusion, its identification with Brownian motion, or the heat-kernel asymptotics) on the DHL cluster is being solved.
  2. [§2] Notation for the various dimensions (degree, resistance, walk) is introduced piecemeal; a consolidated table or definition block in §2 would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below, indicating the revisions we intend to make.

read point-by-point responses

  1. Referee: [§4, Theorem 4.3] §4, Theorem 4.3 and the heat-kernel estimates in §3.2: the on-diagonal bound p_{2n}(x,x) ∼ n^{-d_deg/d_w} is asserted to hold uniformly in both locally finite and locally infinite EIGS regimes, but the argument for the scale-free case (unbounded degrees) does not appear to supply a uniform control on maximal degree or logarithmic growth that would guarantee the required volume-doubling and Hölder constants. Without this, an extra slowly-varying factor can enter the estimate and propagate into the tightness and identification steps for the GHPS-Skorokhod limit, undermining the claim that the limit is the resistance diffusion.

    Authors: We appreciate the referee highlighting the need for explicit uniformity in the scale-free regime. The EIGS definition is based on a finite collection of edge types with fixed iteration rules; this structure directly implies that the maximal degree at iteration level k grows at most exponentially with a rate determined solely by the edge-type data, yielding uniform volume-doubling and Hölder constants independent of scale and of the particular infinite realization. While this control is used implicitly in the proofs of the heat-kernel bounds and the subsequent tightness argument in Theorem 4.3, we acknowledge that it should be stated explicitly. In the revision we will add a short lemma (new Lemma 3.4) that extracts the uniform constants from the finite-type assumption and inserts the resulting bound into the on-diagonal estimate. This addition will confirm that no extra slowly-varying factor appears and that the GHPS-Skorokhod limit remains the resistance diffusion. revision: partial

  2. Referee: [§6] §6 (application to DHL percolation cluster): the resolution of the Hambly-Kumagai open problem is presented as a corollary of the general EIGS theory, yet the manuscript does not explicitly verify that the percolation cluster satisfies the structural hypotheses of an EIGS (in particular the edge-iteration rules and the resulting degree dimension) or isolate which precise statement from the 2010 paper is now settled.

    Authors: We agree that the connection to the DHL percolation cluster should be made fully explicit. In the revised §6 we will insert a dedicated verification subsection showing that the infinite DHL percolation cluster arises as an EIGS generated by two edge types under the standard percolation iteration rules on the diamond hierarchical lattice. We will compute the resulting degree dimension explicitly and confirm that the structural hypotheses of the general theory are satisfied. We will also state precisely that the corollary settles the existence of the scaling limit of the simple random walk on the infinite percolation cluster to the diffusion process associated with the resistance metric, which is the open question posed by Hambly and Kumagai (Commun. Math. Phys. 2010). revision: yes

Circularity Check

0 steps flagged

No circularity: convergence proof is self-contained and references external open problem

full rationale

The paper claims a convergence result for rescaled random walks on Edge Iterated Graph Systems to a limiting diffusion in the Gromov-Hausdorff-Prokhorov-Skorokhod topology, with the degree dimension serving as a correction in on-diagonal heat-kernel estimates for both finite and infinite regimes, and solves the DHL percolation cluster problem from Hambly-Kumagai. No quoted equations or steps reduce the claimed derivation to a self-definition, fitted input renamed as prediction, or load-bearing self-citation chain. The argument is presented as a direct proof relying on graph structural properties and external references, remaining independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The framework rests on the definition of Edge Iterated Graph Systems and standard convergence topologies; no explicit free parameters or new physical entities are introduced in the abstract.

axioms (1)
  • standard math Gromov-Hausdorff-Prokhorov-Skorokhod topology is the appropriate setting for convergence of rescaled random walks on graphs
    Invoked for the main convergence statement in the abstract.
invented entities (1)
  • Edge Iterated Graph Systems (EIGS) no independent evidence
    purpose: Generate a broad class of fractal graphs supporting the random-walk analysis
    Presented as the central construction enabling the unified treatment of finite and infinite degree regimes.

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