Tuesday, October 23, 2012

Lecture 6

This lecture marks the half way point of the course. We celebrated by proving George Polya's famous theorem (1921): that symmetric random walk is recurrent on $\mathbb{Z}$ and $\mathbb{Z}^2$, but transient on $\mathbb{Z}^d$ ($d>2$). In fact, we only proved transience only for $d=3$, but it is easy to see that this implies transience for all $d>2$ (Example Sheet 2, #2).

I mentioned and recommended Polya's famous little book "How to Solve It". I also quoted this story (from A. Motter):

While in Switzerland Polya loved to take afternoon walks in the local garden. One day he met a young couple also walking and chose another path. He continued to do this yet he met the same couple six more times as he strolled in the garden. He mentioned to his wife: how could it be possible to meet them so many times when he randomly chose different paths through the garden?

You can use the results in today's lecture to do Example Sheet 1 #15. It is easy to see from the hint that if we know that symmetric random walk on $\mathbb{Z}^2$ is recurrent then symmetric random walk on a honeycomb lattice must also be recurrent. But proving the converse is harder.

In fact, one can show that random walk on every sort of sensibly-drawn planar graph is recurrent. This includes the honeycomb lattice, and even graphs as strange as the Penrose tiling (which gives us a non-periodic graph). To prove this we must decide what we mean by a "sensibly-drawn planar graph" and then find a way to embed any such graph in in some other graph on which we know random walk is recurrent. You can learn more about the details of this in Peter Doyle and Laurie Snell's Random walks and electric networks, 2006 (freely available to download). The title of this book hints at a connection between random walks and electrical networks. I will explain the connection in Lecture in 12.

The material in Section 6.5 (Feasibility of wind instruments) was written after a short email correspondence with Peter Doyle. The story about cities which beggar their neighbours is my own fanciful invention, but it makes that same point without needing to brush up on the theory of fluid dynamics. It is topical in the light of recent controversy about companies taking advantage of the low corporation tax rates in Ireland and Luxembourg.