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Uniqueness of the critical long-range percolation metrics
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abstract
In this work, we study the random metric for the critical long-range percolation on $\mathbb{Z}^d$. A recent work by B\"aumler [3] implies the subsequential scaling limit, and our main contribution is to prove that the subsequential limit is uniquely characterized by a natural list of axioms. Our proof method is hugely inspired by recent works of Gwynne and Miller [42], and Ding and Gwynne [25] on the uniqueness of Liouville quantum gravity metrics.
Forward citations
Cited by 3 Pith papers
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Critical long-range percolation II: Low effective dimension
In the long-range low-dimensional regime of percolation, the cluster volume tail and k-point functions are determined up to constants, yielding the hyperscaling identities delta=(d+alpha)/(d-alpha) and d_f=(d+alpha)/2.
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Critical long-range percolation I: High effective dimension
In the regime d > min{6, 3alpha}, critical long-range percolation clusters have an n^{-1/2} volume tail and integrated superprocess scaling limits, switching from super-Levy (alpha < 2) to super-Brownian (alpha >= 2);...
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Critical long-range percolation III: The upper critical dimension
For long-range percolation with d=3α<6, the critical volume tail is ~(log n)^{1/4}/√n, the critical two-point function is ~||x-y||^{-d+α}, and superprocess scaling limits hold with explicit logarithmic corrections.
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