#
Percolation on transitive graphs

May 5 to May 9, 2008
at the

American Institute of Mathematics,
Palo Alto, California

organized by

Gabor Pete and Mark Sapir

## Original Announcement

This workshop will be devoted to
percolation on transitive graphs, most importantly, on Cayley graphs
of finitely generated infinite groups. Geometric properties of Cayley
graphs often turn out to have counterparts in the probabilistic
world, and vice versa, but the translations between the different
viewpoints are not always trivial. The aim of this workshop is to
bring together people working in geometric group theory, probability
and dynamics to learn from each other about the relevant techniques
in these fields and thus generate new momentum to solve some of the
persistent open problems.
Some specific problems we
would like address:

- Universality of critical percolation behavior: the conjectures regarding
*p*_{c}<1,
*p*_{c}<*p*_{u}, *p*_{u}<1, quasi-isometry invariance, critical exponents, the role of
asymptotic cones. Could scaling limits be described analogously to SLE?
- What is the relation between lace expansion and the triangle condition of
percolation theory, the rapid decay property, and hyperbolicity of groups?
- On what groups is renormalization possible?
- Relation of group theoretic properties (Kazhdan's property T, L
^{2} and bounded
cohomology, cost of groups, finite presentability) to percolation.
- Survival of geometric and random walk properties under percolation.
- The role of unimodularity in percolation.

## Material from the workshop

A list of participants.
The workshop schedule.

A report on the workshop activities.