  Many real-world networks exhibit the so-called small-world phenomenon: their
typical distances are much smaller than their sizes. One mathematical model for
this phenomenon is a long-range percolation graph on a $d$-dimensional box
$\{0, 1, \cdots, N\}^d$, in which edges are independently added between
far-away sites with probability falling off as a power of the Euclidean
distance. A natural question is how the resulting diameter of the box of size
$N$, measured in graph-theoretical distance, scales with $N$. This question has
been intensely studied in the past and the answer depends on the exponent $s$
in the connection probabilities. In this work we focus on the critical regime
$s = d$ studied earlier in a work by Coppersmith, Gamarnik, and Sviridenko and
improve the bounds obtained there to a sharp leading-order asymptotic, by
exploiting the high degree of concentration due to the large amount of
independence in the model.
