Transcription of Percolation Theory - MIT
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Percolation Theory - MIT
Percolation Theory Dr. Kim Christensen Blackett Laboratory Imperial College London Prince Consort Road SW7 2BW London United Kingdom October 9, 2002. Aim The aim of the Percolation Theory course is to provide a challenging and stimulating introduction to a selection of topics within modern theoretical condensed matter physics. Percolation Theory is the simplest model displaying a phase transition. The analytic solutions to 1d and mean-field Percolation are presented. While Percolation cannot be solved exactly for intermediate dimensions, the model enables the reader to become familiar with important concepts such as fractals, scaling, and renormalisation group Theory in a very intuitive way. The text is accompanied by exercises with solutions and visual interactive simulations for the Percolation Theory model to allow the readers to experience the behaviour, in the spirit seeing is be- lieving . The animations can be downloaded via the URL I greatly appriciate the suggestions and comments provided by Nicholas Moloney and Ole Peters without whom, the text would have been incomprehensible and flooded with mistakes.
Percolation theory deals with the numbers and properties of the clusters formed when sites are occupied with probability p, see Fig. (1.1). Figure 1.1: Percolation in 2dsquare lattice of linear size L= 5. Sites are occupied with probability p. In the lattice above, we have one cluster of size 7, a cluster of size 3 and two clusters of size 1
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