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 p

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.

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 p

__p_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