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Heat kernel estimates on book-like graphs

Emily Dautenhahn, Laurent Saloff-Coste

Abstract

In this paper, we prove two-sided heat kernel estimates on what we call "book-like" graphs. These are graphs consisting of pieces that satisfy the parabolic Harnack inequality that are glued together in a sufficiently nice way over a possibly infinite set of vertices. The prototypical example is gluing a copy of the square four-dimensional lattice $\mathbb{Z}^4,$ a copy of $\mathbb{Z}^5$, and a copy of $\mathbb{Z}^6$ by identifying their $x_1$-axes and taking the lazy simple random walk on this glued graph. Our results are flexible enough to handle perturbations of this example, for instance by adding diagonals to one of the lattices or a few extra vertices/edges.

Heat kernel estimates on book-like graphs

Abstract

In this paper, we prove two-sided heat kernel estimates on what we call "book-like" graphs. These are graphs consisting of pieces that satisfy the parabolic Harnack inequality that are glued together in a sufficiently nice way over a possibly infinite set of vertices. The prototypical example is gluing a copy of the square four-dimensional lattice a copy of , and a copy of by identifying their -axes and taking the lazy simple random walk on this glued graph. Our results are flexible enough to handle perturbations of this example, for instance by adding diagonals to one of the lattices or a few extra vertices/edges.
Paper Structure (17 sections, 12 theorems, 86 equations, 2 figures)

This paper contains 17 sections, 12 theorems, 86 equations, 2 figures.

Table of Contents

  1. introduction
  2. Notation and Set-up
  3. Random Walk Structure on Graphs
  4. Random walks on subgraphs
  5. Properties of Graphs
  6. Book-like Graphs
  7. Gluing graphs
  8. Cutting graphs
  9. Hypotheses on the spine
  10. Key hypotheses
  11. Gluing heat kernels
  12. Abstract gluing estimates
  13. Estimates of quantities in Theorem \ref{['gluing_thm']}
  14. General heat kernel estimates
  15. Example: The Spine is finite
  16. ...and 2 more sections

Key Result

Theorem 2.1

Given $(\Gamma, \pi, \mu)$ (or $(\Gamma, \mathcal{K}, \pi)$) where $\Gamma$ has controlled weights and $\mathcal{K}$ is uniformly lazy, the following are equivalent:

Figures (2)

  • Figure 1: Adding extra dimensions to the figure as necessary, glue copies of $\mathbb{Z}^4, \mathbb{Z}^5, \mathbb{Z}^6$ by identifying their $x_1$-axes: all three vertices of each color shown are the same vertex.
  • Figure 2: Begin with the lazy simple random walk on $\mathbb{Z}^2$ in the top left. Vertex weights are shown, and all edges have weight one. We cut it apart by removing the set of vertices where $y=0$, leaving us the two graphs with trailing edges on the top right. (The "removed" $\Gamma_0$ vertices are treated as a disconnected set all with weight zero, and are not shown.) By capping the trailing edges with vertices, adding back edges between these vertices, and taking the Neumann random walk, we obtain the bottom left pair of graphs with given vertex weights and all edge weights one. If these bottom left graphs are glued back together, one does not get precisely $\mathbb{Z}^2$ with lazy SRW. However, consider instead cutting apart the graphs where weights are assigned as in the bottom right, with all unlabeled edges having weight one. These bottom right graphs can be glued back together into $\mathbb{Z}^2$ with the lazy simple random walk.

Theorems & Definitions (57)

  • Definition 2.1: Edge and Vertex Weights
  • Definition 2.2: Markov kernel on a graph
  • Definition 2.3: Heat kernel on a graph
  • Definition 2.4: Controlled weights
  • Definition 2.5: The notation $\approx$
  • Definition 2.6: Uniformly lazy
  • Definition 2.7: Lazy simple random walk
  • Remark 2.1
  • Definition 2.8: Neumann Markov kernel on a subgraph
  • Definition 2.9: Dirichlet Markov kernel on a subgraph
  • ...and 47 more