Planar maps, random walks and the circle packing theorem, Fall 2017
Monday 16-17, Kaplun 118
Wednesday 11-13, Kaplun 324
The circle packing theorem states that any planar graph can be drawn as the tangency graph of a circle packing. This method of drawing graphs has some interesting applications in the study of random walks on the underlying graph, in particular about questions of recurrence or transience of the walk (i.e., does the walk return to the origin infinitely often almost surely?). We will see from scratch how this manifests and study its uses from random planar maps.
- The circle packing theorem from wikipedia.
- The circle packing theorem is a discrete analogue of the Riemann mapping theorem, see section 2 of Rohde's paper.
- Introduction to electric networks and their connection with random walks can be found in chapters 8 and 9 of Peres's St. Flour lecture notes.