Multiplicative Noise and Second Order Phase Transitions

The scale-free distribution of cluster sizes in continuous phase transitions is linked to the law of proportional effect. A numerical study of a two-dimensional Ising model suggests that a cluster size undergoes a multiplicative birth-death process. At the transition the ratio between birth and death rates approaches unity for large clusters, and the resulting steady state shows a power-law behavior. The percolation dynamic, on the other hand, yields a geometric phase transition without ergodicity breaking, where large-scale merging and splitting of clusters dominate the distribution. Instead of short-range birth-death jumps, the percolation transition is characterized by Leacutevi flights along the cluster-size axis.