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We study the long range percolation model on $\mathbb{Z}$ where sites $i$ and$j$ are connected with probability $\beta |i-j|^{-s}$. Graph distances are nowwell understood for all exponents $s$ except in the case $s=2$ where the modelexhibits non-trivial self-similar scaling. Establishing a conjecture ofBenjamini and Berger \cite{BenBer:01}, we prove that the typical distance fromsite 0 to $n$ grows as a power law $n^{\theta(\beta)}$ up to a multiplicativeconstant for some exponent $0<\theta(\beta)<1$ as does the diameter of thegraph on a box of length $n$.

Distances in critical long range percolation