Abstract:

Percolation theory is a subject in the interface between the areas of Probability and Mathematical Physics. It has the appealing feature of presenting a number of problems which can be stated in relatively simple terms, are generally of relevance to Physics, but present real challenges to mathematicians. The basic model can be described as follows. Suppose that $G$ is an infinite connected graph, and that each one of its edges is deleted independently of the others with probability $1-p$. What can one say about the infinite connected components of the resulting subgraph of $G$ (these are called infinite clusters). In particular, how many infinite clusters are there? It turns out that for various graphs there are critical values of $p$ separating regimes in which the answer to this question is different. Since its introduction in the late 1950's, and until quite recently, percolation was mostly studied on graphs which are Euclidean lattices (the motivation being the structure of crystals). Since the mid 1990's it became clear that the study of percolation on more general graphs leads to a wealth of new results and questions. This talk will present some of this.