# Lazy random walks on the torus via Dirichlet forms

One of the simple example graphs I’ve used in some of my research on gossip algorithms has been the 2-dimensional torus with $n$ vertices, which looks like a $\sqrt{n} \times \sqrt{n}$ grid with the top and left edges wrapped around to connect with the bottom and right edges. Every vertex has 4 neighbors. Now imagine a very lazy random walk on this graph in which a random walker moves from vertex $u$ to one of its neighbors with probability $1/n$. It’s “well known” that this random walk takes around $n^2$ steps to mix. That is, if $P$ is the matrix of transition probabilities then

$T_{r} = \frac{1}{1 - \lambda_2(P)} \approx n^2$

Here $\lambda_2(P)$ is the second largest eigenvalue of $P$ and the relaxation time $T_{r}$ is the inverse of the spectral gap of the matrix. One way of characterizing $T_{r}$ for reversible Markov chains is via the Dirichlet form. For a function $f : V \to \mathbb{R}$ on the states of the chain define the Dirichlet form $D(f,f)$ by

$D(f,f) = \frac{1}{2} \sum_{u,v \in V} \pi_u P_{uv} (f(u) - f(v))^2$

In our example the stationary distribution $pi_u = 1/n$ and $P_{uv} = 1/n$ for all edges in the graph. We write $f \perp \pi$ if

$\sum_{v \in V} \pi_v f(v) = 0$

Define a norm associated with $\pi$ via

$\|f\|^2 = \sum_{v \in V} \pi_i f(v)^2$

Then the characterization is

$T_{r} = \sup_{f \perp \pi, f \ne 0} \frac{ \|f\|^2 }{ D(f,f) }$

One question I asked myself today was whether it was “easy” to see what $f$ you should choose in the grid example to get the scaling of $n^2$. Here’s one choice that gives the correct scaling. We’ll set $f$ to be constant on each column. Assume without loss of generality that $\sqrt{n}$ is divisible by 4 and set $m = \sqrt{n}/4$. The values for $f$ on the columns will be like two triangles:

$\{\frac{1}{n}, \frac{2}{n}, \ldots, \frac{m}{n}, \frac{m}{n}, \frac{m-1}{n}, \ldots, \frac{1}{n}, \frac{-1}{n}, \frac{-2}{n}, \ldots, \frac{-m}{n}, \frac{-m}{n}, \frac{-m+1}{n}, \ldots, \frac{-1}{n} \}$

Now we can evaluate the norm, noting that there are $\sqrt{n}$ vertices per column:

$\|f\|^2 = 4 \sum_{i =1}^{m} \frac{1}{n} \sqrt{n} \frac{i^2}{n^2} = c \frac{1}{n}$

This is because the sum of the first $m$ squares scales like $m^3$ and $m = \sqrt{n}/4$. Now turning to the Dirichlet form, note that each difference between columns is at most $2/n$ and there are fewer than $n$ edges for which $f(u) \ne f(v)$. Thus:

$D(f,f) \le \frac{1}{2} n \frac{1}{n} \frac{1}{n} \frac{4}{n^2} = c' \frac{1}{n^3}$

Taking the ratio gives the lower bound of $T_r \ge c''/n^2$ as desired.

The first $f$ I tried was just equal to +1 on the first half of the columns and -1 on the second half of the columns. This ends up giving a suboptimal bound, because the norm $\|f\|^2 = 1$ but in the denominator we get $2 \sqrt{n}$ positive terms. The key is to make all the differences $(f(u) - f(v))^2$ in the denominator small while keeping the average of $\|f\|^2$ large enough. Even though you sum over $n$ small differences in the denominator, it stays small enough to pay for the $\|f\|^2 = c/n$ in the numerator.

While doing this calculation, I noticed that the book Markov Chains and Mixing Times is also online — it makes a handy reference and is a little easier to use than my old go-to, the Aldous-Fill book.