Speaker

:

Dr. Manjunath Krishnapur

Subject Area

:

Mathematics

Venue

:

Lecture Hall - I, Dept of Mathematics

Time

:

4.00 pm

Date  

:

July 29,2008 (Tuesday)

Title

:

Conformal invariance in probability and statistical physics

This is an expository talk whose aim will be to give an introduction to SLE (Schramm-Loewner evolution), discovered by Oded Schramm in 2000 to describe many critical statistical mechanical systems. We start by recalling classical facts concerning convergence of random walk on Z^2 to Brownian motion, and the conformal invariance property of planar Brownian motion. Then we introduce another model of random paths in Z^2, known as the self-avoiding random walk and ask the question (open for fifty years!) of existence of a continuous scaling limit. We use this discrete model to motivate what is called 'conformal markov property'. Then we show how Schramm magically used Loewner's differential equation (1923) for conformal maps of slit domains to classify all possible scaling limits satisfying the conformal markov property into a one-parameter family of random continuous paths! If time permits we shall see some more discrete models whose limits are SLE.