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.
Uniqueness of the critical long-range percolation metrics
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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.
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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.