240161 Percolation theory (V) (WiSe 2018/2019)

Percolation theory deals with the connectivity properties of
large networks from which a certain fraction of the nodes or edges is
randomly removed. Percolation is one of the simplest models in probability
theory which exhibits what is known as critical phenomena, meaning that
there is a natural parameter in the model at which the behavior of the
system drastically changes.

The first part of the course gives a general introduction and treats several
classical results, including the fundamental theorems of Harris, Kesten and
Menshikov, the uniqueness of the infinite cluster and a proof that the
critical probability for bond percolation on the square lattice is 1/2. Next
we turn to more recent results as Smirnov's Theorem on the conformal
invariance of the critical percolation. Finally, we discuss current research
and open problems, including the question of absence of percolation at the
critical point.

Contents: Introduction (definition of percolation, critical phenomenon,
etc.), Basic tools (FKG and BK inequalities, Russo's formula), Bond
percolation on Z^2 (Harris' Theorem, Kesten's Theorem), Exponential decay
(Menshikov's Theorem), Uniqueness of the infinite cluster and critical
probabilities, Conformal invariance (Smirnov's Theorem), Optional topics.

Bibliography

G. Grimmett: Percolation 2ed, Springer 1999; B. Bollobas, O.
Riordan: Percolation, CUP 2006.