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

Consider a long-range percolation model on Z^d where the probability that an edge {x; y} 2 Z^d × Z^d is open is proportional to ║x-y║₂ ^-d-α for some α > 0 and where d > 3 min{2; α }. We prove that in this case the one-arm exponent equals ½ min{4; α}. 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.