Random Walk on a Mobius Band

As the first post of this blog, I should say something about the title "Random Walk", and more importantly, the little horizontal "8" following it (my poor attempt at HTML coding of a Mobius Band). Why Mobius Band (or "Mobius Strip")? And, what IS a Mobius Band? According to Wikipedia.org, it is a one-sided surface, in the sense that you can walk on both sides of it without ever meeting an edge. Imagine an ant on a flat piece of paper - to go from one side of the paper to the other side it must make a U-turn over an edge. So, how is a one-sided surface at all possible? Cut a paper strip 10-inch long and 1-inch across, give it a half-twist and attach the two ends together - you have a Mobius Band! The ant can now go on and on over both sides of the strip and never come across an edge (see Escher's famous depiction above).

As to "Why Mobius Band" - I don't know about you, but happens many times with me that, after my thought process has taken off on a random excursion, it goes over a wide range of different issues and often many sides of the same issue, but eventually comes back to the original topic that triggered the thought in the first place. However, even though the trajectory is zigzag and the issues are unrelated to one another, yet there is the continuity of an underlying thread that binds one thought to its immediate predecessor (much like the thought of a fishing trip triggering another thought on today's lunch menu). A statistical physicist would probably call it an example of a "Markov process", in which the current state is determined solely by the preceding state, and not the ones before that. But I digress: my point is, this underlying continuity resembles to me the absence of an abrupt "edge", and your thought can wander away and flip around and still return to the starting point without ever making a U-turn, as if it is walking on a Mobius Band of its own.