Topological Solitons Seminar: Exact correlations in two-dimensional critical percolation
1 May 2002 00:00 in CM221
"Percolation has been studied by physicists and mathematicians for many decades, and in many forms. It describes a phase transition in porous matter between macroscopically permeable (or conducting) and impermeable (or isolating) phases. The archetypical model is simply a completely uncorrelated mixture of permeable and impermeable elements. In physical systems these elements are pores or grains, and in models they are typically lattice sites (site percolation) or edges (bond percolation). The bond percolation model can be seen as a Q-state Potts model in the Q=1 limit.
In this lecture I will consider this same critical square bond percolation model on an L by infinity strip with a variety of boundary conditions, periodic and otherwise. When the strip is cut into two half-infinite strips, any two sites on the cut may or may not be connected to each other via the pores in one of the half strips. These connections collectively for all sites on the cut may be called a connectivity configuration (CC). Each of these CC occurs with a characteristic probability or weight. For instance in the two most probable CC all sites are mutually connected, or none of them are.
It turns out that these probabilities have surprising properties. For instance, each one is an integer multiple of the weight of the least probable CC. A very detailed but ill-understood connection has emerged of these probabilities with the number of Alternating Sign Matrices, a problem in combinatorics which has received much attention recently. In the lecture this connection will be presented and many other observations on the critical percolation probabilities will be given. The results are all in the form of conjectures, as none of these observations have been proved to date.
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