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

Consider a long-range percolation model on ${\mathbb{Z}}^d$ where the probability that an edge $\{x,y\}\in {\mathbb{Z}}^d\times {\mathbb{Z}}^d$ is open is proportional to $\mathrm{\parallel }x-y{\mathrm{\parallel }}_2^{-d-\alpha }$ for some $\alpha >0$ and where $d > 3 \min{2,\alpha}$\mathrm{.}Weprovethatinthiscasetheone-armexponentequals$ \min{4,\alpha}/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 $\alpha$ when $\alpha =4$.