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

TL;DR

This paper studies random walks on fractal graphs generated by Edge Iterated Graph Systems, proving their convergence to Brownian motion and solving an open problem on DHL percolation clusters.

Contribution

It establishes the convergence of rescaled random walks to diffusion processes on EIGS fractal graphs and addresses an open problem in percolation cluster analysis.

Findings

Rescaled random walks converge to Brownian motion when resistance dimension is positive.

Identifies degree dimension as key for heat-kernel estimates.

Provides a unified framework for locally finite and infinite regimes.

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].