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Published September 27, 2021 | Submitted
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The critical two-point function for long-range percolation on the hierarchical lattice


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.

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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August 20, 2023
October 23, 2023