Exact results for percolation thresholds, enclosed-area distribution functions and correlation functions in percolation

Series: 
Stochastics Seminar
Tuesday, March 1, 2011 - 16:05
1 hour (actually 50 minutes)
Location: 
Skiles 006
,  
Michigan Center for Theoretical Physics, Department of Chemical Engineering, University of Michigan
Various exact results in two-dimensional percolation are presented. A method for finding exact thresholds for a wide variety of systems, which greatly expands previously known exactly solvable systems to such new lattices as "martini" and generalized "bowtie" lattices, is given. The size distribution is written in a Zipf's-law form in terms of the enclosed- area distribution, and the coefficient can be written in terms of the the number of hulls crossing a cylinder.  Additional properties of hull walks (equivalent to some kinds of trajectories) are given.  Finally, some ratios of correlation functions are shown to be universal, with a functional form that can be found exactly from conformal field theory.