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Skip to: ## Non-mathematical description
Your house is made of bricks. For several reasons - price, insulation, weight - those bricks aren't solid pieces of rock - they're filled with tiny holes. Whoever makes the bricks has to be quite sure that when it rains, water can't seep through those holes - otherwise your house would get pretty wet inside. Percolation models ask questions like: how many holes can I put in my bricks (otherwise said, how cheap and lightweight can I make my bricks) without the inside of people's houses getting wet? ## Basic mathematical description
Take the 2-dimensional square lattice Z^2. For each edge ij, flip a fair coin. If the coin comes up heads, colour the edge black. If it's tails, leave the edge white. (We can think of the black bits as holes in the brick.) One of the most famous results in probability theory is that the fair coin is critical here: if we had a slightly higer probability of heads, then there would be an infinite black cluster somewhere (in the brick terminology, there would be a good chance of having a hole running right through our brick). Click the image below to see a simulation: here I've coloured five vertices in different colours, and their components take their colour (with some colours taking precedence over others if two coloured vertices are in the same component). ## My research
Actually I don't have any research in this area. I can tell you a little bit about the questions above though. Yes, there are exceptional times. On the square lattice we don't know exactly how many there are. Well, there are infinitely many of course, but we don't know the Hausdorff dimension of the set. But if we work on the hexagonal lattice, the Hausdorff dimension of the set of exceptional times is 31/36. And if we make sense of the question properly, then "what does the infinite black cluster look like at exceptional times" can be answered by "it looks like a black cluster conditioned to have diameter at least r, when r goes to infinity" or something like that. ## Going further
Of course there's no reason why we have to consider percolation only on lattices. In fact another of the most famous models in probability, the Erdös-Rényi random graph, is percolation on the complete graph with n vertices. If we choose the probability of heads to be p(n), then something interesting happens around p(n) = 1/n. If p(n) is a bit bigger than 1/n, then as n tends to infinity we see a "giant component": there are a constant times n vertices that are all connected to each other via black paths. If p(n) is a bit smaller than 1/n, then the largest component we see is of order log(n). If p(n) is very close to 1/n, then the largest component has size n^(2/3). The image below shows some realisations of the Erdös-Rényi random graph. The component containing vertex 1 is coloured in red, unless it also contains vertex 2, in which case it's green. Click the image for a simulation of the dynamical process in action. |