## Olivetti Club

Suppose that cracks randomly form in the rock layer above a cave. What are the chances that the water from a flood is able to make its way from the surface to the cavern below? Of course, the answer will depend on the volume of rock lying between the surface and the cave, but also on how likely the cracks are to form in the first place: a mile of sandstone should have more cracks than a mile of diamond, after all. Bernoulli Percolation is a simple mathematical model for this scenario: we model the rock with a graph (the d-dimensional integer lattice) and assume that the cracks form independently and with identical probability at each edge. It is a fascinating example of a system that exhibits critical phenomena: as we vary a parameter (in this case, the probability that a crack forms), the qualitative behavior of the system will eventually undergo a sudden, massive shift! I'll discuss these qualitative changes and demonstrate some of the tools used to study models like this.