Percolation and the Random Cluster Model: Combinatorial and Algorithmic Problems

In 1961 Harry Frisch, John Hammersley and I [13] carried out what were in those days massive Monte Carlo experiments attempting to determine the critical percolation probabilities of the various standard lattices. The constraints at that time were, as today, machine induced. The programmes were written in machine code on a computer which was the size of a large room with less power than a modern day calculator. Today the situation has radically changed. Several of these critical probabilities which we were trying to estimate are now known exactly. However the problems posed then have been replaced by problems of just as much charm and seeming intractability and it is some of these that I shall address in these lectures.