Probability Theory , Phase Transition

G.R. Grimmett

1996, v.2, Issue 1, 51-68

ABSTRACT

The theory of random spatial processes incorporates a probabilistic approach to phase transitions. Many systems exhibiting critical phenomena arise in applied science, and much beautiful mathematics has been discovered in modelling them. Thumbnail sketches are provided of a variety of models, including percolation, fractal percolation, and Ising, Potts, and random-cluster models. The emphasis is upon recent progress obtained by the use of probabilistic methods. These notes are intended as a somewhat telegraphic account of selected parts of a currently major sub-area of probability theory. Fuller versions of the topics referred to here may be found in the references. No serious at attempt has been made to include a complete bibliography, but many key references are listed.

Keywords: phase transition,random graph,Gibbs state,branching process,percolation,fractal percolation,Ising model,Potts model,random-cluster model

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