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• W2035265484 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 $$alpha $$ 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 $$alpha $$ ." @default.
• W2035265484 created "2016-06-24" @default.
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• W2035265484 date "2015-04-04" @default.
• W2035265484 modified "2023-09-25" @default.
• W2035265484 title "Phase transition in loop percolation" @default.
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• W2035265484 doi "https://doi.org/10.1007/s00440-015-0624-x" @default.
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