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

Ziyu Neroli

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arXiv · 2603.13798

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

Ziyu Neroli

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Abstract

This paper investigates random walks and diffusion limits on a broad class of fractals generated by Edge Iterated Graph Systems (EIGS), a generalisation of hierarchical lattices. 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 diamond hierarchical lattice (DHL) percolation cluster posed by Hambly and Kumagai [Commun. Math. Phys. 295 (2010), 29--69]; this suggests that the Alexander--Orbach-type conjecture fails for the natural Brownian motion on the cluster. Sections 2 to 4 have been formalised in Lean.

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Ziyu Neroli. 2026-03-14. Iterated Graph Systems (I): random walks and diffusion limits. https://arxiv.org/abs/2603.13798

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