Answer Happy

5.) Percolation (18 pts.) Consider a 2-dimensional square lattice (N < N). Starting from an empty lattice, sites are subsequently occupied randomly, where n is the number of occupied states at given moment. (a) Define the occupation probability p and the notion of a spanning cluster. How many different spanning clusters can coexist in 2-D? (b) Define the notion of the percolation transition and the critical probability pe(N). What are the minimal and maximal values for pe(N) for N=2 and N=3? (c) Sketch the procedure and a graph for how to extract pe in the infinite-size limit. Define the critical exponent 3 associated with the occupation probability, F, in the spanning cluster above pc, and sketch F as a function of p.