3:30 p.m., Friday
Gregory F. Lawler
Department of Mathematics
Universality and conformal invariance
in two dimensional statistical physics
There are a number of two dimensional
lattice models in statistical physics
(percolation, polymers or self-avoiding walks (SAW),
nonintersecting random walks, uniform spanning
trees or loop-erased walk (LEW), Ising model)
which have been conjectured to have continuum limits
at the critical temperature that
are in some sense conformally invariant. Exact
values of critical exponents have been predicted
(nonrigorously) using conformal field theory.
This colloquium will discuss recent work
with Oded Schramm and Wendelin Werner
on these conjectures. The work focuses
on those models which are expected to have a certain
``restriction property'' in the limit (this includes
SAW, and nonintersecting random walks
but not LEW or Ising model).
*It is shown that the exponents for all conformally
invariant models with the restriction property can be
given in terms of the intersection exponents for
random walk/Brownian motion.
*Another process, the stochastic Loewner equation (SLE),
introduced recently by Schramm, has the restriction
property for a particular parameter value.
*The exponents for the SLE can be calculated exactly.
By universality, we establish rigorously the
exponents for Brownian motion, which in turn proves
the fractal properties of a Brownian path.
While we cannot prove anything about the continuum limit
of percolation and SAW, the results do shed light about what
kind of limits should be expected for these processes.