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

Ziyu Neroli

Abstract

This paper investigates random walks and diffusion limits on a broad class of fractal graphs generated by Edge Iterated Graph Systems (EIGS). We study random walks on combinatorial limit graphs and establish the connections among several dimensions, including the Einstein relation. Building on this, we prove that the rescaled simple random walks converge in the Gromov-Hausdorff-Prokhorov-Skorokhod topology to the limiting diffusion, which further coincides with Brownian motion when the resistance dimension is positive. Moreover, we use the degree dimension to unify the on-diagonal heat-kernel estimates in the locally finite and locally infinite (scale-free) regimes. Finally, we solve the open problem on the quenched resistance exponent for the DHL percolation cluster left in [27].

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

Abstract

This paper investigates random walks and diffusion limits on a broad class of fractal graphs generated by Edge Iterated Graph Systems (EIGS). We study random walks on combinatorial limit graphs and establish the connections among several dimensions, including the Einstein relation. Building on this, we prove that the rescaled simple random walks converge in the Gromov-Hausdorff-Prokhorov-Skorokhod topology to the limiting diffusion, which further coincides with Brownian motion when the resistance dimension is positive. Moreover, we use the degree dimension to unify the on-diagonal heat-kernel estimates in the locally finite and locally infinite (scale-free) regimes. Finally, we solve the open problem on the quenched resistance exponent for the DHL percolation cluster left in [27].
Paper Structure (14 sections, 53 theorems, 278 equations, 12 figures, 1 table)

This paper contains 14 sections, 53 theorems, 278 equations, 12 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Edge iterated graph systems
  4. Main results
  5. Random walks
  6. Resistance renormalisation
  7. Walk dimension
  8. Diffusion limits and Brownian motion
  9. Dirichlet form and diffusion limits
  10. Resistance-form Brownian motion
  11. Heat kernel for the diffusion limit
  12. Solution to the open problem in
  13. Quenched unit-resistance exponent
  14. Heuristic application: Brownian spectral dimensions for the critical percolation cluster of the DHL

Key Result

Lemma 2.3

$\Psi:\mathbb R_{\ge 0}^K\to \mathbb R_{\ge 0}^K$ satisfies:

Figures (12)

  • Figure 1: Example of EIGS: diamond hierarchical lattice (DHL)
  • Figure 2: An example: Canonical Xi graph (in $\mathbb Z^2$)
  • Figure 3: An example of two-coloured EIGS
  • Figure 4: $(u,v)$-flower
  • Figure 5: A sample trace of a $10^6$-step simple random walk on the level-6 approximation $\Xi^6$ of the canonical Xi graph, drawn in its natural embedding in $\mathbb Z^2$. The underlying graph is shown in grey and the trace in red.
  • ...and 7 more figures

Theorems & Definitions (120)

  • Definition 1.1
  • Definition 2.1
  • Example 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4: Renormalisation map
  • proof
  • Remark 2.5
  • Lemma 2.6
  • proof
  • ...and 110 more