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Published September 27, 2021 | Accepted Version
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Power-law bounds for critical long-range percolation below the upper-critical dimension


We study long-range Bernoulli percolation on ℤd in which each two vertices x and y are connected by an edge with probability 1−exp(−β‖x−y‖−d−α). It is a theorem of Noam Berger (CMP, 2002) that if 0<α

Additional Information

Dedicated to Harry Kesten, November 19, 1931 - March 29, 2019. We thank Jonathan Hermon for his careful reading of an earlier version of this manuscript, and thank Gordon Slade for helpful discussions on the physics literature. We also thank the anonymous referee for their helpful comments and corrections.

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Accepted Version - 2008.11197.pdf


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August 19, 2023
October 23, 2023