Some Results for Range of Random Walk on Graph with Spectral Dimension Two

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 Erdös (in: Proceedings of Second Berkeley symposium on mathematical statistics and probability, University of California Press, California, 1951). Our proof does not depend on spatial 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.

Appendix A. Intersection of Two Independent Random Walks on Graph

In this section, we consider intersections of two independent random walks starting at a common point. This case is relatively easier to handle than Sect. 3, but on the other hand the arguments in this section are similar to Sect. 3.

Proposition A.1

Let $S^{(1)}$ and $S^{(2)}$ be two independent simple random walks on G. Let $P^{x,y}$ be the joint law of $S^{(1)}$ and $S^{(2)}$ whose starting points are x and y, respectively. Then, there exists a positive constant C such that for every $\epsilon > 0$ and every $x \in V(G)$,

$$\begin{aligned} P^{x,x}\left( \left| \{S^{(1)}_0, \dots , S^{(1)}_n\} \cap \{S^{(2)}_0, \dots , S^{(2)}_n\} \right| \ge \frac{\epsilon n}{\log n}\right) \le \frac{C (\log \log n)^2}{\epsilon ^2 (\log n)^{2}} \end{aligned}$$

As in [36], the estimate in Proposition A.1 is attributed to obtaining an upper bound for the expectation of the intersection.

Proposition A.2

There exists a positive constant C such that for every $x \in V(G)$ and $n \ge 2$,

$$\begin{aligned} E^{x,x}\left[ \left| \{S^{(1)}_0, \dots , S^{(1)}_n\} \cap \{S^{(2)}_0, \dots , S^{(2)}_n\} \right| \right] \le \frac{C n \log \log n}{(\log n)^2}\end{aligned}$$

For the case that $G = {\mathbb {Z}}^2$, we have a better estimate (see [36, Theorem 5.1]). We conjecture that $\log \log n$ in the numerator of the right-hand side of the above inequality could be replaced with 1. We use estimates for hitting time distributions on graphs (cf. [4]).

Proof

By (1.3) and Lemma 3.3, we have that

$$\begin{aligned}&E^{x,x}\left[ \left| \{S^{(1)}_0, \dots , S^{(1)}_n\} \cap \{S^{(2)}_0, \dots , S^{(2)}_n\} \right| \right] = \sum _{y} P^x (T_y \le n)^2 \\&\quad \le \left| B\left( x, \left( \frac{n}{(\log n)^2}\right) ^{1/d}\right) \right| + \sup _{y \notin B(x, (\frac{n}{(\log n)^2})^{1/d})} P^x (T_y \le n) \sum _{y} P^x (T_y \le n)\\&\quad = O\left( \frac{n}{(\log n)^2}\right) + O\left( \frac{\log \log n}{\log n}\right) E^x [R_n]. \end{aligned}$$

By this and (3.3), we have the assertion. $\square$

The following lemma is related with the moment method.

Lemma 7.13

For every $p \in {\mathbb {N}}$,

$$\begin{aligned}&\sup _{x \in V(G)} E^{x,x} \left[ \left| \{S^{(1)}_0, \dots , S^{(1)}_n\} \cap \{S^{(2)}_0, \dots , S^{(2)}_n\} \right| ^p \right] \\&\quad \le (p!)^2 \left( \sup _{x \in V(G)} E^{x,x} \left[ \left| \{S^{(1)}_0, \dots , S^{(1)}_n\} \cap \{S^{(2)}_0, \dots , S^{(2)}_n\} \right| \right] \right) ^p. \end{aligned}$$

Theorem A.1 easily follows from Proposition A.2 and Lemma 7.13. The proof of Lemma 7.13 is same as in the proof of [37, Lemma 3.1], so we omit the proof.