### ABSTRACT -- Wendelin Werner

Random planar curves and conformal invariance

Understanding the asymptotic behaviour when *N* goes to infinity of the
number *a(N)* of self-avoiding curves
of length *N* on the square lattice **Z** x **Z**
is an tantilising open problem.
Theoretical physicists (Nienhuis, Cardy, Duplantier etc.) have
given various predictions concerning the existence
and value of critical exponents for some
two-dimensional systems in statistical physics
(such as self-avoiding walks, critical percoaltion, intersections
of simple random walk) using considerations related to several branches
of mathematics (probability theory,
complex variables, representation theory of infinite-dimensional Lie
algebras).
For instance, it is expected that *a(N)* grows asymptotically like *c*^{N} N^{11/32}
for some lattice-dependent constant *c*.

The aim of this lecture is to give an introduction to the subject
and to present some mathematical progress obtained
in joint work with Greg
Lawler and Oded Schramm. In particular, we will briefly describe
the proofs of the conjectures concerning
intersections of simple random walks and Brownian motions and of
Mandelbrot's conjecture on the fractal dimension of the outer frontier
of a planar Brownian path ("its Hausdorff dimension is *4/3*").

Part of this talk will be based on the review paper
downloadable as
gzipped postscript
or
postscript.

Click here for pdf version of abstract.

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Last modified: February 28, 2001

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