Basic Notions: Ori Gurel Gurevich (HUJI) - On Smirnov's proof of conformal invariance of critical percolation

Abstract:

Let G be an infinite connected graph. For each vertex of G we decide
randomly and independently: with probability p we paint it blue and
with probability 1-p we paint it yellow. Now, consider the subgraph of
blue vertices: does it contain an infinite connected component?

There is a critical probability p_c(G), such that if p>p_c then almost
surely there is a blue infinite connected component and if pp_c or p<p_c.

We will focus on planar graphs, specifically on the triangular
lattice. For the triangular lattice, p_c=1/2 and we will consider the
critical case, p=p_c. We think of the lattice as embedded in the
plane, with distances between adjacent vertices being epsilon.

In his breakthrough paper, Smirnov proved that when we let epsilon go
to 0, the limit behaviour is conformally invariant, that is, the limit
behaviour in some domain, when transformed by a conformal map to a
different domain, is the same as the limit behaviour in the second
domain.

This is a basic notions talk. All concepts will be explained. The only
requirement is basic knowledge of probability and complex analysis.

Date: 

Thu, 02/03/2017 - 16:00 to 17:00

Location: 

Manchester Building, Lecture Hall 2