Continuous phase transitions on Galton-Watson trees
When does a system undergo a continuous phase transition, and when does a system undergo a first-order (i.e., discontinuous) phase transition? This is the question in some of the most central problems in discrete probability and statistical physics, like whether bond percolation occurs at criticality on the lattice in dimensions 3 to 10. Though physicists have many nonrigorous thoughts about this, not much is known in general. We look at the question for branching process events satisfying recursive properties. For example, let T_1 be the event that a Galton-Watson tree is infinite, and let T_2 be the event that it contains an infinite binary tree starting at the root. The event T_1 holds if and only if T_1 holds for at least one of the trees initiated by children of the root, and T_2 holds if and only if T_2 holds for at least two of these trees. The probability of T_1 has a continuous phase transition, increasing from 0 when the mean of the tree’s child distribution increases above 1. On the other hand, the probability of T_2 has a first-order phase transition, jumping discontinuously to a nonzero value at criticality. We give some explanation of why, explaining the connection between the recursive property satisfied by the event and the phase transition the event experiences.