The investigator has collaborated recently with Benjamini, Peres and Schramm in the area of probabilistic processes on graphs whose distributions are invariant under automorphisms of the graph. One of the areas concerns uniform spanning trees, where an enormous number of new questions have opened up because of the interconnections of the field with other areas of probability such as random walks and Dirichlet functions. The investigator proposes to work on several such questions. Another area concerns percolation, both classical Bernoulli percolation and more general invariant percolation. A large number of questions in this area are also proposed.
The earliest result directly related to this investigator's work was discovered by the physicist Kirchhoff in 1847, who showed that electrical network problems are precisely related to certain probabilistic properties of special subnetworks called spanning trees. The connections between these topics have flourished in recent years and have also found links to other fundamental probabilistic processes known as random walks and percolation. Random walks model many sorts of random change, such as prices on the stock market. Percolation is a probabilistic model of diffusion through a porous medium, such as groundwater seeping through soil. The proposed work involves various connections among these and other fields. The investigator aims to further the fundamental understanding that is part of basic research.
