Hutchcroft, Tom (2022) Sharp hierarchical upper bounds on the critical two-point function for long-range percolation on ℤᵈ. Journal of Mathematical Physics, 63 (11). Art. No. 113301. ISSN 0022-2488. doi:10.1063/5.0088450. https://resolver.caltech.edu/CaltechAUTHORS:20221129-370786800.2
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Abstract
Consider long-range Bernoulli percolation on ℤᵈ in which we connect each pair of distinct points x and y by an edge with probability 1 − exp(−β‖x − y‖^(−d−α)), where α > 0 is fixed and β ⩾ 0 is a parameter. We prove that if 0 < α < d, then the critical two-point function satisfies (1/|Λ_r|)∑_(xϵΛ_(r))P_(β_(c))(0 ↔ x) ≤ r^(−d+a) for every r ⩾ 1, where Λ_r = [−r,r]ᵈ ∩ ℤᵈ. In other words, the critical two-point function on ℤᵈ is always bounded above on average by the critical two-point function on the hierarchical lattice. This upper bound is believed to be sharp for values of α strictly below the crossover value α_(c)(d), where the values of several critical exponents for long-range percolation on ℤᵈ and the hierarchical lattice are believed to be equal.
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Additional Information: | We thank Philip Easo, Emmanuel Michta, Gordon Slade, and the anonymous referee for helpful comments on earlier versions of the manuscript. | ||||||
Issue or Number: | 11 | ||||||
DOI: | 10.1063/5.0088450 | ||||||
Record Number: | CaltechAUTHORS:20221129-370786800.2 | ||||||
Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20221129-370786800.2 | ||||||
Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||||
ID Code: | 118141 | ||||||
Collection: | CaltechAUTHORS | ||||||
Deposited By: | Research Services Depository | ||||||
Deposited On: | 22 Dec 2022 16:59 | ||||||
Last Modified: | 22 Dec 2022 16:59 |
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