The one-arm exponent for mean-field long-range percolation

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Consider a long-range percolation model on Zᵈ where the probability that an edge {x,y} ∈ Zᵈ × Zᵈ is open is proportional to x-y₂⁻ᵈ⁻α for some α>0 and where d > 3 {2,α}. We prove that in this case the one-arm exponent equals {4,α}/2. We also prove that the maximal displacement for critical branching random walk scales with the same exponent. This establishes that both models undergo a phase transition in the parameter α when α=4.