# Some results for range of random walk on graph with spectral dimension two

## TL;DR

This paper investigates the behavior of the range of simple random walks on graphs with spectral dimension two, establishing strong laws of large numbers and stability results, with applications to lamplighter random walks.

## Contribution

It provides a new proof of the strong law of large numbers for the range on the 2D lattice and extends results to fractal graphs and finite modifications.

## Key findings

- Strong law of large numbers holds under a uniform condition.
- The scaled expectations of the range are stable under finite modifications.
- Constructs a recurrent graph where the uniform condition holds but expectations fluctuate.

## Abstract

We consider the range of the simple random walk on graphs with spectral dimension two. We give a form of strong law of large numbers under a certain uniform condition, which is satisfied by not only the square integer lattice but also a class of fractal graphs. Our results imply the strong law of large numbers on the square integer lattice established by Dvoretzky and Erd\"os (1951). Our proof does not depend on spacial homogeneity of space and gives a new proof of the strong law of large numbers on the lattice. We also show that the behavior of appropriately scaled expectations of the range is stable with respect to every "finite modification" of the two-dimensional integer lattice, and furthermore we construct a recurrent graph such that the uniform condition holds but the scaled expectations fluctuate. As an application, we establish a form of law of the iterated logarithms for lamplighter random walks in the case that the spectral dimension of the underlying graph is two.