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 [LeJ12] and [LL12]. 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 [LL12] 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 $\mathbb{Z}^d$ ($d\geq 3$) 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 $d\geq 5$, 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 $\alpha$. [LeJ12] Y. Le Jan, Amas de lacets markoviens, C. R. Math. Acad. Sci. Paris 350 (2012), no.13-14, 643-646. [LL12] Y. Le Jan and S. Lemaire, Markovian loop clusters on graphs, arXiv.org:1211.0300