Special topics course proposal
Title: Schramm-Loewner Evolutions and Two-Dimensional Statistical Physics
Instructor: Tom Kennedy
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.
Text: We will roughly follow Greg Lawler's book "Conformally invariant
processes in the plane" which may be downloaded from his website.
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.