Project: How useful of simple random walk to statistical mechanics?

[zzzchong]

Simple random walk has been believed to be super-simple, and tons of rigorous results have been dig out. In statistical mechanics, many models have their graph representation and are more or less related to various "walks" models, especially in very high dimensions. However, rigorous for these models on Z^d are very rare. How we can make useful of simple random walk to understand the behaviour of these statistical mechanical models?