ABSTRACT


The percolation model describes the formation of connectivity in random network systems. The scaling theory of percolation will be reviewed and reformulated in a universal form emphasizing the universal fractal nature of this critical phenomenon. Some recent rigorous and conjectured mathematical results that relate to crossing and wrapping probabilities will be summarized, including work by Langlands, Aizenman, Cardy, Duplantier, and Smirnov. Some new numerical observations suggesting an elegant exact result for the number of percolating clusters in strip geometries will be presented. To verify numerically the conjectured formulas for multiple crossing probabilities given by Cardy, Duplantier and Aizenmen, a new rareevent simulation technique (due to Grassberger and Ziff) will be described. Finally, some interesting aspects of percolation on smallworld networks  combination of a random graph and a regular lattice  will be discussed. Return to Applied Math Colloquium home page 