Tom Hutchcroft: Critical behaviour in hierarchical percolation (Caltech)

Statistical mechanics models undergoing a phase transition often exhibit rich, fractal-like behaviour at their critical points, which are described in part by critical exponents - the indices governing the power-law growth or decay of various quantities of interest. These exponents are expected to depend on the dimension but not on the microscopic details of the model such as the choice of lattice. After much progress over the last thirty years we now understand two-dimensional and high-dimensional models rather well, but intermediate dimensions such as three remain mysterious. I will discuss these issues in the context of hierarchical percolation, a simplified version of percolation on Z^d for which, in forthcoming work, I establish a fairly good description of the critical behaviour in all dimensions.