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Percolation TheoryDr. Kim ChristensenBlackett LaboratoryImperial College LondonPrince Consort RoadSW7 2BW LondonUnited KingdomOctober 9, 2002AimThe aim of the percolation theory course is to provide a challenging and stimulating introductionto a selection of topics within modern theoretical condensed matter physics.Percolation theory is the simplest model displaying a phase transition. The analytic solutionsto 1d and mean-field percolation are presented. While percolation cannot be solved exactly forintermediate dimensions, the model enables the reader to become familiar with important conceptssuch 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 thepercolation 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 http://www.cmth.ph.ic.ac.uk/kim/cmth/I greatly appriciate the suggestions and comments provided by Nicholas Moloney and Ole Peterswithout whom, the text would have been incomprehensible and flooded with mistakes. However,if you still are able to find any misprints, misspellings and mistakes in the notes, I would be verygrateful if you would report those to [email protected] Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Percolation in 1d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Percolation in the Bethe Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.5 Cluster Number Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.6 Cluster Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.6.1 Cluster Radius and Fractal Dimension . . . . . . . . . . . . . . . . . . . . . . 221.6.2 Finite Boxing of Percolating Clusters . . . . . . . . . . . . . . . . . . . . . . . 241.6.3 Mass of the Percolating Cluster . . . . . . . . . . . . . . . . . . . . . . . . . . 251.7 Fractals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.8 Finite-size scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.9 Real space renormalisation in percolation theory . . . . . . . . . . . . . . . . . . . . 311.9.1 Renormalisation group transformation in 1d. . . . . . . . . . . . . . . . . . . 341.9.2 Renormalisation group transformation on 2d triangular lattice. . . . . . . . . 351.9.3 Renormalisation group transformation on 2d square lattice of bond percolation. 361.9.4 Why is the renormalisation group transformation not exact? . . . . . . . . . 3721.1 IntroductionPercolation theory is the simplest not exactly solved model displaying a phase transition. Often,the insight into the percolation theory problem facilitates the understanding of many other physicalsystems. Moreover, the concept of fractals, which is intimately related to the percolation theoryproblem, is of general interest as it pops up more or less everywhere in Nature. The knowledge ofpercolation, fractals, and scaling are of immense importance theoretically in such diverse fields asbiology, physics, and geophysics and also of practical importance in e.g. oil recovery. We will begingently by developing a basic understanding of percolation theory, providing a natural introductionto the concept of scaling and renormalisation group theory.1.2 PreliminariesLet P (A) denote the probability for an event A and P (A1∩A2) the joint probability for event A1and A2.Definition 1 Two events A1and A2are independent ⇔ P (A1∩ A2) = P (A1)P (A2).Definition 2 More generally, we define n ≥ 3 events A1, A2, . . . , Anto be mutually independentif P (A1∩A2∩ ··· ∩An) = P (A1)P (A2) ···P (An) and if any subcollection containing at least twobut fewer than n events are mutually independent.Let each site in a lattice be occupied at random with probability p, that is, each site is occupied(with probability p) or empty (with probability 1−p) independent of the status (empty or occupied)of any of the other sites in the lattice. We call p the occupation probability or the concentration.Definition 3 A cluster is a group of nearest neighbouring occupied sites.Percolation theory deals with the numbers and properties of the clusters formed when sites areoccupied with probability p, see Fig. (1.1).Figure 1.1: Percolation in 2d square lattice of linear size L = 5. Sites are occupied with probabilityp. In the lattice above, we have one cluster of size 7, a cluster of size 3 and two clusters of size 1(isolated sites).Definition 4 The cluster number ns(p) denotes the number of s-clusters per lattice site.The (average) number of clusters of size s in a hypercubic lattice of linear size L is Ldns(p), d beingthe dimensionality of the lattice. Defining the cluster number per lattice site as opposed to thetotal number of s-clusters in the lattice ensures that the quantity will be independent of the latticesize L.3For finite lattices L < ∞, it is intuitively clear, that if the occupation probability p is small,there is only a very tiny chance of having a cluster percolating between two opposite boundaries(i.e., in 2d, from top-to-bottom or from left-to-right). For p close to 1, we almost certainly willhave a cluster percolating through the system. In Fig. 1.2, sites in 2d square lattices are occupiedat random with increasing occupation probability p. The occupied sites are shown in gray whilethe sites belonging to the largest cluster are shown in black. Unoccupied sites are white. Note thatfor p ≈ 0.59, a percolating cluster appears for the first time.Figure 1.2: Percolation in 2d square lattices with system size L ×L = 150 ×150. Occupation prob-ability p = 0.45, 0.55, 0.59, 0.65, and 0.75, respectively. Notice, that the largest cluster percolatesthrough the lattice from top to bottom in this example when p ≥ 0.59.Definition …


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