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Critical percolation and random spanning forests, with connections to geometric group theory and conformal mapping
Colloquium| Speaker: | Yuval Peres, UC Berkeley |
| Location: | 693 Kerr |
| Start time: | Mon, Oct 7 2002, 4:10PM |
Description
Percolation was introduced by mathematicians in the
1950's, and later studied intensively by physicists, as it is
the simplest model exhibiting a phase transition.
I'll describe the current status of the conjecture that
critical percolation on a Cayley graph admits no infinite
clusters (still open for the 3D lattice!), and sketch the
recent proof of this conjecture in the non-amenable case.
This conjecture is closely related to the structure of
"Minimal Spanning Forests". These natural yet mysterious objects
are expected to undergo a "qualitative transition" every eight
dimensions. Finally, I will describe the conformal invariance
conjecture for planar critical percolation, recently proved for
the triangular lattice but still open for all others.