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On the Number of Entangled Clusters

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Abstract

We prove that the number of entangled clusters with N edges in the simple cubic lattice grows exponentially in N. This answers an open question posed by Grimmett and Holroyd (Proc. Lond. Math. Soc. 81:485–512, 2000). Our result has immediate implications for entanglement percolation: we obtain an improved rigorous lower bound on the critical probability, and we prove that the radius of the entangled component of the origin has exponentially decaying tail when p is small.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, Ontario, M3J 1P3, Canada

    Mahshid Atapour & Neal Madras

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  1. Mahshid Atapour
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  2. Neal Madras
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Correspondence to Neal Madras.

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Atapour, M., Madras, N. On the Number of Entangled Clusters. J Stat Phys 139, 1–26 (2010). https://doi.org/10.1007/s10955-010-9941-8

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