Phase transition in loop percolation

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 (alpha ) and (kappa ). The non-negative parameter (alpha ) measures the amount of loops, and (kappa ) plays the role of killing on vertices penalizing ((kappa >0)) or favoring ((kappa <0)) appearance of large loops. It was shown in Le Jan (Ill J Math 57(2):525–558, 2013) that for any fixed (kappa ) and large enough (alpha ), there exists an infinite cluster in the loop percolation on ({mathbb {Z}}^d). In the present article, we show a non-trivial phase transition on the integer lattice ({mathbb {Z}}^d) ((dge 3)) for (kappa =0). More precisely, we show that there is no loop percolation for (kappa =0) and (alpha ) small enough. Interestingly, we observe a critical like behavior on the whole sub-critical domain of (alpha ), namely, for (kappa =0) and any sub-critical value of (alpha ), the probability of one-arm event decays at most polynomially. For (dge 5), we prove that there exists a non-trivial threshold for the finiteness of the expected cluster size. For (alpha ) 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 (d=3) 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 (alpha ).