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Subcritical Connectivity and Some Exact Tail Exponents in High Dimensional Percolation

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Abstract

In high dimensional percolation at parameter \(p < p_c\), the one-arm probability \(\pi _p(n)\) is known to decay exponentially on scale \((p_c - p)^{-1/2}\). We show upper and lower bounds on the same exponential scale for the ratio \(\pi _p(n) / \pi _{p_c}(n)\), establishing a form of a hypothesis of scaling theory. As part of our study, we provide sharp estimates (with matching upper and lower bounds) for several quantities of interest at the critical probability \(p_c\). These include the tail behavior of volumes of, and chemical distances within, spanning clusters, along with the scaling of the two-point function at “mesoscopic distance” from the boundary of half-spaces. As a corollary, we obtain the tightness of the number of spanning clusters of a diameter n box on scale \(n^{d-6}\); this result complements a lower bound of Aizenman (Nucl Phys B 485(3):551–582, 1997).

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No data was generated in the course of the research described in this manuscript.

Notes

  1. Here we use Aizenman’s [1] definition of “spanning cluster”; other natural definitions of this term exist.

  2. The letter “E” in the abbreviation “EREG” refers to “expectation”. Compare our definition to that of regularity appearing in [30, Section 4].

  3. The letter “S” in the abbreviation “SREG” stands for “shell”. The regularity condition is restricted to a shell to allow us to decouple portions of the cluster.

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Acknowledgements

The authors thank Akira Sakai for helpful discussions about the problem addressed in Theorem 6. The authors also thank two anonymous referees for extensive and helpful comments on an earlier version of this manuscript.

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Authors and Affiliations

  1. Department of Mathematics, City University of New York, City College and Graduate Center, New York, USA

    Shirshendu Chatterjee & Jack Hanson

  2. Department of Mathematics, Cornell University, Ithaca, USA

    Philippe Sosoe

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  1. Shirshendu Chatterjee
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Correspondence to Jack Hanson.

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The research of S.C. is supported in part by NSF grant DMS-2154564. The research of J.H. is supported in part by NSF grant DMS-1954257. The research of P.S. is supported in part by NSF grant DMS-1811093.

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Chatterjee, S., Hanson, J. & Sosoe, P. Subcritical Connectivity and Some Exact Tail Exponents in High Dimensional Percolation. Commun. Math. Phys. 403, 83–153 (2023). https://doi.org/10.1007/s00220-023-04759-w

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