Dynamical critical site percolation on the triangular lattice has exceptional times

You need to know: Graph, infinite graph, connected subgraph of a graph, basic probability theory, exponential distribution with parameter .

Background: The triangular grid is the (infinite) graph G whose vertex set is given by , and two vertices are connected by an edge if and only if the distance between them is . Assume that each vertex of G, independently of other vertices, (i) is coloured black or white at time , with probability for each colour, (ii) generate a random variable according to exponential distribution with parameter , wait for time , and switch the colour, and (iii) repeat part (ii) again and again. This is called the dynamical critical site percolation on the triangular grid G. If, at some moment , graph G has an infinite connected subgraph consisting on only white vertices, we say that percolation occurs. Let be the set of all times at which percolation occurs.

The Theorem: On 29th April 2005, Oded Schramm and Jeffrey Steif submitted to arxiv a paper in which they proved that, with probability , the set  is non-empty.

Short context: Site percolation is an important model in mathematics for studying processes such as diffusion. It is studied for various grids, such as triangular, square, etc. In the triangular grid model described above, it is known that at any fixed , the probability that percolation occurs at time is . The Theorem states, however, that, with probability , there is a non-empty set of exceptional times at which percolation occurs.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 10.3 of this book for an accessible description of the Theorem.

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