Spring 2002, MWF 12:30-1:20
One of the main goals of both probability theory and statistical
physics is to understand the asymptotic behavior of random systems when the
number of microscopic random inputs goes to infinity. In order to understand these
asymptotic behaviors, one often attempts to find a continuous model and prove
the convergence to this continuous model.
Simple random walk on any lattice in Rn converges to a Brownian motion in the scaling limit is
the simplest and most important example.
However,
establishing such a scaling limit is not always easy, in fact it is quite
difficult for many stochastic and physical models. Whereas mathematicians have
been struggling to prove existence of such limits (and even finding meaningful definitions!)
for some systems such as percolation, loop-erased random walk and self avoiding walk, physicists went far ahead and
provided many results (called ``predictions'' by numerous mathematicians, because
of the lack of the rigorous mathematical proof). The huge gap between these
predictions on the one hand and the mathematically proven facts on the other
hand was recently narrowed: A new tool, the Stochastic Loewner Evolution SLE
introduced by Oded Schramm in 1999, has been used to prove numerous predictions. In this 2-quarter sequential
course, we will explain the main
results of this fast growing new field.
The problems and results in this area belong to the realm of Probability, but there is a significant amount of complex analysis (in particular conformal mapping) involved. Prerequisites are the first year real and complex analysis graduate courses, and basic concepts of probability theory. In order to keep background minimal, we will not assume familiarity with Brownian motion, but rather prove some of its basic properties (conformal invariance, recurrence). We will develop the necessary amount of stochastic calculus (partly without proof). We will discuss SLE (definition, basic properties, phases, Hausdorff dimension) and hope to cover Smirnov's Theorem (critical percolation converges to SLE(6))) at the end of spring quarter. It is expected that a continuing course will be offered in the Autumn quarter.
There is no textbook. We will provide a list
of relevant papers at the beginning of the Spring quarter and will distribute
some course notes as time goes.
Prerequisites: The
first year real and complex analysis graduate courses, and basic concepts of probability
theory.