Mathematical Problems in Percolation and Ising Models

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9803598 Wu The aim of this research is the study of random spatial systems on the finite dimensional hypercubic lattices and infinite dimensional hyperbolic lattices. Three topics concerning percolation, Ising (ferromagnetic) models, and contact processes will be studied. The first topic concerns percolation and Ising models on hyperbolic lattices. Although Ising models and percolation 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 a few results have been rigorously proved, many statements suggested by numerical studies are to be proved; and many more are to be explored. The second topic concerns a roughening transition of (independent and dependent) percolation on the hypercubic lattices with dimensions larger than or equal to three. This is the analogy of a roughening transition of ferromagnetic Ising models. The third topic is about a phase transition of models with low-dimensional inhomogeneity. These models include percolation, Ising ferromagnetic systems and contact processes. This research by an investigator at an undergraduate institution involves the study of percolation and interacting particle systems. These are probabilistic models, which have been useful in understanding physical systems. Further understanding of these probabilistic models can be helpful to physicists.