Avoidance Coupling

I took a read through this fun paper that appeared on ArXiV a few months ago:

Avoidance Coupling
Omer Angel, Alexander E. Holroyd, James Martin, David B. Wilson, Peter Winkler
arXiv:1112.3304v1 [math.PR]

Typically when you think of coupling arguments, you want to create two (or more) copies of a Markov chain such that they meet up quickly. A coupling of Markov chains with transition matrix $P$ is a sequence of pairs of $\{(U_t, V_t)\}$ such that $\{U_t\}$ and $\{V_t\}$ are Markov chains with transition matrix $P$:

$\mathbb{P}( U_{t+1} = j | U_t = i, V_t = i') = P(i,j)$
$\mathbb{P}( V_{t+1} = j' | U_t = i, V_t = i') = P(i', j')$

Note that the two chains need not be independent! The idea is that we start $\{U_t\}$ and $\{V_t\}$ at different initial positions and when they “meet” we make them move together. Once they meet, the decisions from random mappings are the same for both chains. The coupling time is $T_c = \min \{t : U_t = V_t \}$. Coupling is used to show fast mixing of Markov chains via theorems like the following, which says that the difference between the distribution of the chain started at time $i$ and the chain started at time $i'$ at time $t$ is upper bounded by the probability of coupling:

Theorem. Let $\{ (U_t,V_t) \}$ be a coupling with $U_0 = i$ and $V_0 = i'$. Then

$\| P^t(i,\cdot) - P^t(i',\cdot) \|_{\mathrm{TV}} \le \mathbb{P}_{(U,V)}\left( T_c > t \right)$

This paper takes an entirely different tack — sometimes you want to start off a bunch of copies of the chain such that they never meet up. This could happen if you are trying to cover more of the space. So you want to arrange a coupling such that the coupling time is huge. There’s a bit of a definitional issue regarding when you declare that two walkers collide (what if they swap places) but they just say “multiple walkers are assumed to take turns in a fixed cyclic order, and again, a collision is deemed to occur exactly if a walker moves to a vertex currently occupied by another. We call a coupling that forbids collisions an avoidance coupling.”

There are a number of results in the paper, but the simplest one to describe is for two walkers on a complete graph $K_n$ (or the complete graph with loops $K_n^{\ast}$.

Theorem 4.1. For any composite $n = ab$, where $a,b > 1$, there exist Markovian avoidance couplings for two walkers on $K_n$ and on $K_n^{\ast}$.

How do they do this? They partition $n$ into $b$ clusters $\{S_i\}$ of size $a$. Let’s call the walkers Alice and Bob. Suppose Alice and Bob are in the same cluster and it’s Bob’s turn to move. Then he chooses uniformly among the vertices in another cluster. If they are in different clusters, he moves uniformly to a vertex in his own cluster (other than his own). Now when it’s Alice’s turn to move, she is always in a different cluster than Bob. She picks a vertex uniformly in Bob’s cluster (other than Bob’s) with probability $\frac{a(b-1)}{ab - 1}$ and a vertex in her own cluster (other than her own) with probability $\frac{a - 1}{ab - 1}$.

So let’s look at Bob’s distribution. The chance that he moves to a particular vertex outside his current cluster is the chance that Alice moved into his cluster times the uniform probability of choosing something outside his cluster:
$\frac{a(b-1)}{ab - 1} \times \frac{a (b-1)} = \frac{1}{ab - 1}$
The chance that he moves to a vertex inside his own cluster is likewise
$\frac{a - 1}{ab - 1} \times \frac{1}{a-1} = \frac{1}{ab - 1}$
So the marginal transitions of Bob are the same as a uniform random walk.

For Alice’s distribution we look at the time reversal of the chain. In this case, Alice’s reverse distribution is Bob’s forward distribution and vice versa, so Alice also looks like a uniform random walk.

There are a number of additional results in the paper, such as:

Theorem 7.1. There exists a Markovian avoidance coupling of $k$ walkers on $K_n^{\ast}$ for any $k \le n/(8 log_2 n)$, and on $K_n$ for any $k \le$n/(56 log_2 n)\$.

Theorem 8.1. No avoidance coupling is possible for $n - 1$ walkers on $K_n^{\ast}$ , for $n \ge 4$.

In addition there are a ton of open questions at the end which are quite interesting. I didn’t mention it here, but there are also interesting questions of the entropy of the coupling — lower entropy implies easier simulation, in a sense.