Hutchcroft, Tom (2021) The critical two-point function for long-range percolation on the hierarchical lattice. . (Unpublished) https://resolver.caltech.edu/CaltechAUTHORS:20210924-202140311
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Abstract
We prove up-to-constants bounds on the two-point function (i.e., point-to-point connection probabilities) for critical long-range percolation on the d-dimensional hierarchical lattice. More precisely, we prove that if we connect each pair of points x and y by an edge with probability 1-exp(-β||x-y||^(-d-α)), where 0 < α < d is fixed and β ≥ 0 is a parameter, then the critical two-point function satisfies P_(β_c)(x ↔ y)||x-y||^(-d+α) for every pair of distinct points x and y. We deduce in particular that the model has mean-field critical behaviour when α < d/3 and does not have mean-field critical behaviour when α > d/3.
Item Type: | Report or Paper (Discussion Paper) | ||||||
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Additional Information: | This research was supported by ERC starting grant 804166 (SPRS). We thank Gordon Slade for helpful comments on a previous version of the manuscript. | ||||||
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Record Number: | CaltechAUTHORS:20210924-202140311 | ||||||
Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20210924-202140311 | ||||||
Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||||
ID Code: | 111038 | ||||||
Collection: | CaltechAUTHORS | ||||||
Deposited By: | George Porter | ||||||
Deposited On: | 27 Sep 2021 15:03 | ||||||
Last Modified: | 27 Sep 2021 15:03 |
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