I am considering the spectral norm of a transition matrix on simple random walk om $\mathbb{Z}^2$ as follows.
Let $S_n$ be a simple random walk on $\mathbb{Z}^2$. Let $D$ be a connected subset of $\mathbb{Z}^2$ and $A$ be the matrix whose entries are the transition probability between the elements of $D$. The matrix $A$ has properties
- A is a strictly doubly sub-stochastic matrix.
- Every entry of A is $0$ or $\frac{1}{4}$.
Let $D$ be a ball with radius $n$ in $\mathbb{Z}^2$, that is $D=D_n= \{z \in \mathbb{Z}^2: |z|<n\}$. Here $|\cdot|$ represent for Euclidian norm in $\mathbb{R}^2$.
For instant with $n=2$ we have $D_2$ has 9 points, and the matrix $A$ as follows
$$A=A_2 =\left(\begin{matrix} 0 & \frac{1}{4} & 0 & \frac{1}{4} & 0 & 0 & 0 & 0 & 0 \\ \frac{1}{4} & 0 & \frac{1}{4} & 0 & \frac{1}{4} & 0 & 0 & 0 & 0 \\ 0 & \frac{1}{4} & 0 & 0 & 0 & \frac{1}{4} & 0 & 0 & 0 \\ \frac{1}{4} & 0 & 0 & 0 & \frac{1}{4} & 0 & \frac{1}{4} & 0 & 0 \\ 0 & \frac{1}{4} & 0 & \frac{1}{4} & 0 & \frac{1}{4} & 0 & \frac{1}{4} & 0 \\ 0 & 0 & \frac{1}{4} & 0 & \frac{1}{4} & 0 & 0 & 0 & \frac{1}{4} \\ 0 & 0 & 0 & \frac{1}{4} & 0 & 0 & 0 & \frac{1}{4} & 0 \\ 0 & 0 & 0 & 0 & \frac{1}{4} & 0 & \frac{1}{4} & 0 & \frac{1}{4} \\ 0 & 0 & 0 & 0 & 0 & \frac{1}{4} & 0 & \frac{1}{4} & 0 \end{matrix}\right).$$
Let $\lambda(D)$ be the spectral radius of A. It is the largest eigenvalue of $A$ since $A$ is symmetric. Using Python, I can guess that if $n$ is large then $\lambda \approx 1$ moreover $$1-\lambda \approx \frac{c}{n^2}.$$
I would like to prove that
If $D_n$ is a ball with radius $n$ of $\mathbb{Z}^2$ then $$1-\lambda \approx \frac{c}{n^2}$$ as $n \to \infty$.
May someone give me some hints or references to prove it? Many thanks.