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

Abstract: 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 $|x-y|_2^{-d-\alpha}$ for some $\alpha >0$ and where $d > 3 \min{2,\alpha}$. We prove that in this case the one-arm exponent equals $ \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 $\alpha =4$.