Recent progress in one-dimensional and high-dimensional statistical mechanical models

In this talk, I will summarize recent progress made in a number of statistical mechanical models, such as the self-avoiding walk and percolation. Self-avoiding walks are a model for polymers, whereas percolation can be used to study the flow of fluids through porous media. Progress has mainly been made in dimension one and high dimensions.

In high dimensions, this progress is mainly due to the use of the lace expansion. In dimension one, the lace expansion can also be used. However, most progress is made using large deviation techniques. This talk will be panoramic, in the sense that it will highlight the new developments, and give a flavour of the proofs rather than the technical details.

The talk is based upon joint work with Takashi Hara, Frank den Hollander, Wolfgang Koenig and Gordon Slade.