Grimmett: The Random-Cluster Model

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“The random-cluster model as studied so far is random in space but not in time. There are a variety of ways of introducing time-dynamics into the model, and some good reasons for so doing. The principal reason is that, in our 3 + 1 dimensional universe, the time-evolution of processes is fundamental. It entails the concepts of equilibrium and convergence, of metastability, and of chaos. A rigorous theory of time-evolution in statistical mechanics is one of the major achievements of modern probability theory with which the names Dobrushin, Spitzer, and Liggett are easily associated.” (p. 222) #Grimmett #Random-ClusterModel #Space #Time #ProbabilityTheory

Grimmett: Percolation

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“Suppose we immerse a large porous stone in a bucket of water. What is the probability that the centre of the stone is wetted? In formulating a simple stochastic model for such a situation, Broadbent and Hammersley (1957) gave birth to the ‚percolation model‘.” (p. 1) #Grimmett #Broadbent #Hammersley #percolation