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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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.