Abstract:

The random cluster measure is a model for random graphs that, apart from being interesting in its own right, allows one to study certain equilibrium distributions in statistical mechanics by geometric means. In particular, the question of uniqueness of equilibrium (Gibbs) states in the Ising/Potts model for ferromagnets becomes a question of connectivity in its random cluster counterpart. On a finite graph, there is a strong link between the random cluster measure and the Tutte polynomial of the underlying graph; a similar analysis in the Ising/Potts model suggests yet another model for random graphs, namely that of random Poisson flows. Aizenmann and others used Poisson flows to prove an important result about correlations in the Ising model, and we explore their methods in this talk. The question is still open if their methods can be extended to more general Potts or random cluster models.