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.
This is a continuation of the course we gave in the Spring quarter of 2002. During the Spring quarter, we have studied the basic properties of Brownian motion and stochastic calculus as well as basics of SLE (including its definition, basic properties, phases, Hausdorff dimension, restriction and universality). See the course notes for details.
In this course, we plan to cover Smirnov's Theorem (critical percolation converges to SLE(6)), SLE8/3 (restriction; SAW), intersection exponents for Brownian motion, properties of SLE (with proofs), convergence of LERW and UST to SLE2 and SLE8, and Hausdorff dimension for Brownian frontier.
There is no textbook. We will provide a list of relevant papers at the beginning of the Autumn quarter and will distribute some course notes as time goes.
Prerequisites:
The first year
real and complex analysis graduate courses, and basic knowledge
of Brownian motion and stochastic calculus.