## Monday, August 16, 2010

### Random walks can explain a lot.

The simple idea of taking seemingly random events and grouping them to create a predictable distribution is genius. I always knew Einstein was awesome. It’s got me thinking about other day-to-day processes which operate around this principle/diffusive behaviour, or don’t as the case may be.
As it had been a rather wet week my first thought was rain (I love rain). The pattern droplets make on concrete seems like the perfect image of a diffusion process at a boundary. Perhaps using the right experimental setup one could measure the permeability of concrete using the diffusion model. Although this may be an interesting concept it probably wouldn’t be very practical and I’m sure road-workers have their own faster, more accurate method. My second thought was about something I also love, Tea. Obviously the teabag is the perfect membrane model and it would be relatively easy to determine its permeability (thus leading to the careful creation of the perfect cup of tea. Delicious!).
After running out of further interesting ideas and a quick google check, it seems random walks are useful for a number of things including population genetics, studying the human brain, the share market and much more (It’s even used in art).

1. Brownian motion and random walks were mentioned in my maths lecture. We were doing a unit on financial mathematics. Apparently they are used a lot in the models economists use to predict stock prices. It is assumed in financial mathematics that the stock market moves at a given time independently of the times before it. It also has structure on many levels (I don't know if it has existed long enough to say it has structure on all levels.)

While it is good to have a way to model Brownian motion, it is not an easy thing to perform calculus on. Things that exhibit a random walk are always continuous (a particle can't jump suddenly and leave a gap in its path) but absolutely no where is it differentiable. Differential equations of this form are called stochastic differential equations, and are even more of mission to solve than partial differential equations.

2. The allusion to the stock market is appropriate - although people always have their own "reasons" for buying & selling, the net motion appears like a random walk.

One important point to make is that people's financial decisions are not independent - they are correlated. You're more/less likely to buy depending on what the people you know are doing.

In the natural sciences (as in economics), correlations between parts of the system are hard to deal with. To make the problem tractable, a "mean field" approximation is often made. This leads to equations that describe the average value of something, expressed as an effective system where the parts are independent. The correlations then appear in equations determining the higher moments of the distribution.

It is interesting to note that there is a trick in quantum statistical physics where one maps the behavior of a correlated system onto a one-body system where the potential becomes a random variable. This is called the Hubbard-Stratonovich transformation. Here is a wikipedia page:

http://en.wikipedia.org/wiki/Hubbard-Stratonovich_transformation

and here are various notes in PDF form: