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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.