PERCOLATION ON REGULAR LATTICES AND SMALL-WORLD NETWORKS

ROBERT M. ZIFF

Michigan Center for Theoretical Physics

Sep 24,
4:15pm

2-105

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 rare-event simulation technique (due to Grassberger
and Ziff) will be described.  Finally, some interesting aspects of
percolation on small-world networks --- combination of a random graph and
a regular lattice --- will be discussed.




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