# Greg Lawler - Virginia Mathematics Lectures - February 12-14, 2020

#### Event start date: Wednesday, February 12, 2020

##### Greg Lawler (University of Chicago)

- Lecture 1 - “
**Random walks: Simple and Self-avoiding**”. Wednesday, February 12, 2020. 5:15-6:15pm, Nau 101
- Lecture 2 - “
**Conformal Invariance and the Two-Dimensional Critical Phenomenon**”. Thursday, February 13, 2020. 5-6pm, Monroe 124
- Lecture 3 - “
**Loop Measures and the Loop-Erased Random Walk**”. Friday, February 14, 2020. 4-5pm, Monroe 124

### Lecture 1. Random walks: Simple and Self-avoiding

The most common model for random behavior is the “drunkard’s walk” where at each time an individual chooses their step from some probability distribution. I will review this and then discuss what happens when one puts some constraints on the walker to try to avoid places already visited. We will see the relationship between the “fractal dimension” of the random path and the ambient dimension in which it lives.

It was predicted by theoretical physicists that lattice models from equilibrium statistical physics “at criticality” in two dimensions have limits that are conformally invariant. There has been an incredible amount of work in the last twenty years making these ideas precise and rigorous and I will survey this work. The starting point was the development of the Schramm-Loewner evolution (SLE) which I will define.

### Lecture 3. Loop Measures and the Loop-Erased Random Walk

This talk will focus on two related models: loop measures and the loop-erased random walk which are closely related to uniform spanning trees and describe some relatively recent work in this area in dimensions two, three, and four.

Virginia Mathematics Lectures archive

**Last updated:** Saturday, December 7, 2019