# Winding Number

### Winding number

Choose a uniform spanning tree T in a 200x200 square grid. Associate to the path from the lower left corner to a vertex x its net number h(x) of turns (also called its winding number), i.e., the number of times it turns left minus the number of times it turns right. Let h(T) be the mean value of h(x) over all vertices x. The scaling limit of the distribution of $(\sqrt\pi/4)\big(h - h(T)\big)$ is the Gaussian free field, as shown by Rick Kenyon (and conjectured in looser form by Itai Benjamini). In the sample shown here, with net turns coded by color and ranging from -7 to +8, the mean winding number is about 0.917. - Russell Lyons