# Mathematics Colloquia and Seminars

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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 |

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.