Phase transition in loop percolation

Abstract

We are interested in the clusters formed by a Poisson ensemble of Markovian loops on infinite graphs. This model was introduced and studied in Le Jan (C R Math Acad Sci Paris 350(13–14):643–646, 2012, Ill J Math 57(2):525–558, 2013). It is a model with long range correlations with two parameters (Formula presented.) and (Formula presented.). The non-negative parameter (Formula presented.) measures the amount of loops, and (Formula presented.) plays the role of killing on vertices penalizing ((Formula presented.)) or favoring ((Formula presented.)) appearance of large loops. It was shown in Le Jan (Ill J Math 57(2):525–558, 2013) that for any fixed (Formula presented.) and large enough (Formula presented.) , there exists an infinite cluster in the loop percolation on (Formula presented.). In the present article, we show a non-trivial phase transition on the integer lattice (Formula presented.) ((Formula presented.)) for (Formula presented.). More precisely, we show that there is no loop percolation for (Formula presented.) and (Formula presented.) small enough. Interestingly, we observe a critical like behavior on the whole sub-critical domain of (Formula presented.) , namely, for (Formula presented.) and any sub-critical value of (Formula presented.) , the probability of one-arm event decays at most polynomially. For (Formula presented.) , we prove that there exists a non-trivial threshold for the finiteness of the expected cluster size. For (Formula presented.) below this threshold, we calculate, up to a constant factor, the decay of the probability of one-arm event, two point function, and the tail distribution of the cluster size. These rates are comparable with the ones obtained from a single large loop and only depend on the dimension. For (Formula presented.) or 4, we give better lower bounds on the decay of the probability of one-arm event, which show importance of small loops for long connections. In addition, we show that the one-arm exponent in dimension 3 depends on the intensity (Formula presented.).