RUI: Some Problems in Percolation and Stochastic Dynamics of Ising Models

The aim of this research is to study several random spatial models on various infinite graphs. Four topics concerning stochastic dynamics of Ising spin systems, the random cluster and Ising ferromagnetic model, and the contact process are proposed to be studied. The first topic concerns stochastic dynamics of an Ising spin system on an infinite lattice where spins evolve according to the usual Glauber dynamics. Some typical questions are: does a spin flip infinitely many times or only finitely many times? what is the probability that a spin has not yet flipped at time t? The second topic is about Ising models on hyperbolic lattices. Although Ising models on the hypercubic lattices have been studied intensively and extensively since they were introduced, these models on hyperbolic lattices have just started to receive attention from physicists and mathematicians. They are found, by both numerical studies and mathematical proofs, to exhibit a phenomenon of multiple phase transitions. Although some results have been rigorously proved, many statements suggested by numerical studies are to be proved, and many more are to be explored. Some GHS type inequalities in the random cluster model and a related question of uniqueness of the random cluster measure are the contents of the third topic. The final topic deals with phase transitions of models with low-dimensional inhomogeneity. These models include percolation, Ising ferromagnetic systems and contact processes.

Models of the sort to be studied in this research arise naturally from physical sciences. Percolation is a probabilistic model of studying flow through a discrete disordered system, such as particles flowing through the filter of a gas mask, or fluid seeping through the interstices of a porous medium, while the contact process can be regarded as modeling the spread of an epidemic through a population.