Schramm-Loewner Evolutions and Two-Dimensional Statistical Physics
Special Topics Course - Spring 2005
Meeting time: MWF 10:00 in Math 501
This course will be an introduction to one of the most spectacular
recent discoveries at the interface of mathematics and physics -
the Schramm-Loewner Evolution (SLE), which brings together ideas from
probability, complex analysis, geometry and physics.
SLE is a one parameter family of stochastic processes that
produce random curves in the plane. The past few years have
shown that these processes decribe the random curves that are
found in a wide variety of two dimensional statistical physics
systems at their critical point. "Critical points" are values
of the parameters of the system with the property that the randomness
of the system, which typically is seen only at microscopic length
scales, produces random structures that can be seen at macroscopic
length scales.
Conformal invariance plays a key idea in the SLE process and its connection
to statistical physics models. Roughly speaking conformal invariance means
that the model is not changed by a map in the plane that preserves angles.
There are lots of such maps in two dimensions, and the rich nature of
this symmetry produced in the 1980's something called
conformal field theory which revolutionized our understanding of
two-dimensional critical phenomena. The results of the past few years on
SLE are advancing our understanding of the geometry of two-dimensional
critical phenomena in an equally exciting fashion.
Some examples of two-dimensional models described by SLE include
the self-avoiding walk, the loop-erased random walk,
interfaces in critical percolation, and the frontier of Brownian motion.
This course will start with an explanation of what all these models are.
Then we will motivate and define the Schramm-Loewner
evolution process, including some background in conformal maps (the Loewner
equation) and stochastic differential equations.
Next we will derive some of the properties of the SLE process. Two
particular properties of SLE only hold for special values of the parameter,
and this fact helps make the connection between particular cases of SLE
and particular physics models. Finally we will take a look at some of
the proofs that particular models are described by SLE.
Prerequisites: SLE involves conformal maps in the plane, Brownian motion
and stochastic differential equations, but I will cover what we need
to know (as does Lawler's book). I will assume an undergraduate
knowledge of complex variables and probability.
If you are planning to take the course and have not officially
enrolled, please do so. The course number is Math529, section 1.
At present there are enough students for the course to run, but
just barely.
Here are some review articles and a book:
- Wendelin Werner's lectures on
"Random Planar Curves and Schramm-Loewner Evolution"
at the St. Flour summer school.
- Notes from a mini-course on
"Conformal Restriction and Related Questions" by Wendelin Werner.
- A draft of Greg Lawler's coming book,
"Conformally invariant processes in the plane" , may be downloaded
from his website.