A sharp leading order asymptotic of the diameter of a long range percolation graph

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.