Illinois Geometry Lab Percolation Theory Authors: Michelle Delcourt Kaiyue Hou Yang Song Zi Wang Faculty Mentor: Kay Kirkpatrick December, 03
Classical percolation theory includes site and bond percolations In a grid structure, each cell is a site and the bond is the edge between cells Two cells sharing an edge are neighbors to each other A path is a sequence of neighboring sites on the grid By assigning each site/bond with a probability p of being occupied/open, we are interested whether we could find a path from the leftmost to the rightmost or from the top to the bottom We have created several 3-dimensional visualizations of percolation models Our project also focuses on the bootstrap percolation process Bootstrap percolation is a model for disease infection: In a grid, if the site is initially occupied independently with a random probability p, recursively other sites will be infected if the number of infected neighbors is not less than a predefined threshold r We are interested in whether the whole grid will be infected eventually We specifically focus on the -dimensional grid -neighbor and 4-neighbor bootstrap percolation processes are relatively simple Although -neighbor bootstrap percolation has been extensively studied, the critical probability for 3-neighbor bootstrap percolation process is still open Site Percolation Imagine filling a box with glass and metal balls placed randomly A natural question is: Does the entire box act like a conductor or an insulator? In other words, is there a spanning cluster, a group of metal balls touching each other that reaches from one side of the box to the other? Consider a random network where vertices are metal balls with probability p and glass balls with probability of ( p) Edges in the network correspond to where balls touch each other We want to know what is the critical probability to make the whole grid behave like conductor on an infinite grid In other words, there is a path of metal balls from top to the bottom or from left to right In figure, blue cells are metal The path that we are interested in does not exist in the first figure but does exist in the second one Figure : 35 35 Grid Site Percolation Example: (i) p = 05: Does not Percolate; (ii) p = 065: Does Percolate
Bond Percolation Imagine pouring a liquid on top of a block of porous material and letting the liquid trickle down Will the liquid be able to travel through an open path from top to bottom? Consider a random network in which an edge is open with probability p, and therefore blocking liquid with probability p Similar to site percolation, we are curious what is the critical probability that there is a path from top to the bottom or from left to right on a infinite grid Figure illustrates this situation Figure : 40 40 Grid Bond Percolation Example: (i) Does not Percolate; (ii) Does Percolate The critical probability for the -dimensional lattice and the Bethe lattice are and (z ) respectively, where z is the number of neighbors of each site in the Bethe lattice In 980, Kesten[5] showed that the critical probability for a -dimensional grid is The intuition behind this critical probability is that if we cannot find the path from the top to the bottom, then there is a path formed by closed edges from left to the right By symmetry, we could estimate that the critical probability is about 3 Bootstrap Percolation The k-dimensional r-neighbor bootstrap percolation process is as follows: Similar to site percolation, each site in a k-dimensional grid is occupied independently with a predefined probability New sites will be infected in each step if the site has at least r neighbors Will the whole grid be infected eventually? We are interested in whether the total probability that the grid is completely infected is greater than The critical threshold of the predefined probability is called critical probability and denoted by p On the -dimensional grid, the critical probability p for the -neighbor bootstrap percolation process is 0 If p > 0, some sites are infected and all the neighbors of the occupied sites will be infected Eventually, the whole grid is infected The -neighbor bootstrap percolation
Figure 3: -neighbor Bootstrap Percolation: (i) Initial Configuration; (ii) Cells with at least two neighbors are marked; (iii) Marked cells are infected has been extensively studied Gravner et al[4] show that the critical probability is π 8logn C(loglogn)3 (logn) 3 p π 8logn for every n N, where C and c are some positive numbers c (logn) 3 4 Improvement for 4-Neighbor Bootstrap Percolation The 4-neighbor bootstrap percolation process is equivalent to whether the initial configuration of the grid contains a uninfected cluster of size If there is such a cluster, then each site in the cluster has an uninfected neighbor Each site in the cluster has at most 3 occupied neighbors Hence, all the cells in the cluster will not be infected Conversely, if there is no such cluster, then all the neighbors of the uninfected cell are occupied and will be infected The exception is sites on the sides Since each site on the side of the grid has at most three neighbors, if some cell is not occupied, then it cannot be infected To simplify the computation, we ignore the cells on the sides Therefore, ( ) n ( p) = n(n ) ; Using Stirling s formula, we get p = 4 ( ) n n p = Θ ( n 4 (n e) n ) 3
This result could be generalized to k-dimensional k-neighbor bootstrap percolation process The critical probability is ( ) p = Θ n k 4 k (n e) n(k ) 5 Improvement for 3-Neighbor Bootstrap Percolation The 3-neighbor bootstrap percolation problem is equivalent to whether the initial configuration contains a cycle of unoccupied sites If there is such a cycle, then each site has two neighbors uninfected In other words, each site has at most two occupied neighbors Hence, cells in the cycle will never be infected Conversely, if there is no such cycle, the structure of the uninfected sites is a forest In each tree of the forest, there is a leaf that has only one uninfected neighbor Thus, the leaf has three unoccupied neighbors and will be infected in the next round After removing a leaf from the tree, we could find new leaves Therefore, all the sites will be infected eventually This equivalence can be generalized to k-dimensional (k )-neighbor bootstrap percolation problem On the -dimensional grid, the upper bond of the critical probability is given by that there is no uninfected size--cluster The lower bound is given by that there is a uninfected cluster From the above two conditions, we have ( ) ( ) n 8 n 4 Θ ( n e) < p < Θ n (n e) n In the future, it would be interesting to refine the result of k-dimensional k-neighbor bootstrap percolation by considering the sites on the sides of the grid, and to refine the result of k-dimensional (k )-neighbor bootstrap percolation problem 6 References József Balogh, Béla Bollobás, Robert Morris Bootstrap Percolation in Three Dimensions, 009 József Balogh, Gabor Pete Random Disease on the Square Grid, 998 Kim Christensen, Percolation Theory, MIT Lecture Notes, 00 Janko Gravner, Alexander E Holroyd, Robert Morris A Sharper Threshold for Bootstrap Percolation in Two Dimensions, 00 Harry Kesten The Critical Probability of Bond Percolation on the Square Lattice Equals /, 980 4