Phase transition in loop percolation (Q267029)

The authors consider a model of loops on an arbitrary graph where the probability of the occurence of an ensemble of loops depends on two parameters: \(\kappa\) (which controls the length of the loops) and \(\alpha\) (which controls the number of loops). They consider the problem of percolation for this loop ensemble when the graph is \(\mathbb{Z}^d\). For \(d \geq 3\), they show the existence of a phase transition in the \(\alpha, \kappa\) parameter space. That is, they show that there is a critical curve \(\kappa = \kappa_c(\alpha)\) such that there exists an infinite cluster on one side of the curve, and only finite-sized clusters on the other side. They also present detailed estimates on the size of the clusters in the sub-critical regime.