Tuesday, October 25, 2011

Lecture 6

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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 Z and Z^2, but transient on 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?
I talked a bit about electrical networks in which each edge of an infinite graph is replaced with a 1 ohm resistor. I am thinking here not only of graphs such as the regular rectangular or honeycomb lattice, but also graphs as strange as the Penrose tiling (which gives us a non-periodic graph). I told you that the resistence between two nodes in such a network tends to infinity (as the nodes are chosen a distance from one another that tends to infinity) if and only if a symmetric random walk on the same graph is recurrent. I will say more about why this is so in Lecture in 12.

Meanwhile, you can apply today's lecture to Example Sheet 1 #15. It is pretty easy to see from the hint that if we know that symmetric random walk on 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. 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 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..

This map shows the sources of visitors to this Markov Chains course web site over the past 4 days (20-23 October). Visitors to the course page are averaging about 45 per day and on this blog page about 15 per day. It seems that about 25% of the hits to the course page are from outside Cambridge.