Heat kernel estimates on book-like graphs

TL;DR

This paper establishes two-sided heat kernel estimates on 'book-like' graphs formed by gluing together lattice pieces satisfying the parabolic Harnack inequality, including perturbations such as added diagonals or extra vertices.

Contribution

It introduces a method to derive heat kernel estimates on complex glued graphs built from well-understood lattice components, extending previous results to more general structures.

Findings

Two-sided heat kernel estimates are proven for book-like graphs.

The estimates are robust under perturbations like added diagonals or extra vertices.

The framework applies to graphs formed by gluing multiple lattice structures.

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.