#### Milestone Year

### 1996

#### Percolation beyond Euclidean Lattices

Classical Percolation theory studies the large scale structure of random
subgraphs of Euclidean Lattices obtained by deleting edges independently at
random, providing a model for the behavior of random media,
and in particular for phase transitions that occur in the model.

This study was extended by Weizmann mathematicians, who initiated in 1996
the study of percolation on general geometries such as the graph
constructed via actions of a groups. They discovered qualitatively
different behavior of the percolation process in these structures, where
the underlying geometry of the structure, such as hyperbolic geometry, plays
a key role. This study lead to the resolution of a couple of geometrical
and group theoretic problems and to fomulating some still open conjectures.

In 1999, the theory was further generalized to arbitrary group invariant
probability measures in contrast to the classical study that was confined
to product measures, e.g. the Ising model. In particular, it led to the
introduction and study of the uniform spanning tree model and various
models of self-interacting random walks.

In the same year, a new notion of noise sensitivity of Boolean functions
was introduced and studied mostly in the context of planar Bernoulli
percolation. Applications include solving the long standing open problem
of establishing "super concentration" of natural random metrics,
meaning that these random behavior is more concentrated near its mean
than the aggregate of random noise.