Grimmett: The Random-Cluster Model


“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, Geoffrey R., The Random-Cluster Model. With 37 Figures. Berlin/Heidelberg: Springer 2006. 377 S., ISBN 978-3-540-32890-2.

Grimmett: Percolation


“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

Grimmett, Geoffrey R., Percolation. With 76 Illustrations. New York: Springer 1989. 296 S., ISBN 978-1-4757-4210-7.