In this lecture we have encountered yet another random process, namely cluster growth. We discussed two models of cluster growth, Eden and DLA, and we found that they lead to strikingly different cluster geometries. We quantified the visual impression by introducing the concept of fractal dimensions.
In the second part of the lecture we discussed a related phenomenon, known as percolation. In its simplest form it is yet another random process, where a lattice is filled up to a certain concentration and the emerging structures are analysed in view of how they extend over the full lattice. While this may sound quite academic here, it should be noted that such simple models are very interesting tools to analyse much more complicated behaviour in the physical world, such as forest fires, oozing of oil through porous material etc.. However, we analysed the way in which percolating clusters emerge, and we found a critical concentration pc. We also discovered that there is a phase transition, most likely of second order, in this seemingly simple model, namely between the percolating and the non-percolating phase. We briefly discussed the behaviour of the model around this phase transition.