Phase transition in loop percolation

TL;DR

This paper investigates phase transitions in loop percolation on infinite lattices, revealing critical behavior and decay rates of connection probabilities, with distinctions based on dimension and parameters, especially at the critical point.

Contribution

It establishes the existence of a non-trivial phase transition for loop percolation on  lattices at  , providing decay estimates and highlighting the role of small loops in connection probabilities.

Findings

No percolation for small  when =0.

Polynomial decay of one-arm event probability in sub-critical regime.

Dimension-dependent decay rates and importance of small loops for long connections.

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