NDLI: Phase transition in loop percolation

Content Provider	Springer Nature Link
Author	Sapozhnikov, Artëm Chang, Yinshan
Copyright Year	2015
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 $${\mathbb {Z}}^d$$ ( $$d\ge 3$$ ) 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 $$d\ge 5$$ , 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 $$\alpha $$ .
Ending Page	1025
Page Count	47
Starting Page	979
File Format	PDF
ISSN	01788051
e-ISSN	14322064
Journal	Probability Theory and Related Fields
Issue Number	3
Volume Number	164
Language	English
Publisher	Springer Berlin Heidelberg
Publisher Date	2015-04-04
Publisher Place	Berlin/Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Mathematical and Computational Biology Statistics for Business/Economics/Mathematical Finance/Insurance Theoretical, Mathematical and Computational Physics Interacting random processes; statistical mechanics type models; percolation theory Probability Theory and Stochastic Processes Operation Research/Decision Theory Quantitative Finance Markov chains (discrete-time Markov processes on discrete state spaces)
Content Type	Text
Resource Type	Article
Subject	Statistics and Probability Analysis Statistics, Probability and Uncertainty

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3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
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