  In this paper, we study graph distances in the geometric random graph models
scale-free percolation SFP, geometric inhomogeneous random graphs GIRG, and
hyperbolic random graphs HRG. Despite the wide success of the models, the
parameter regime in which graph distances are polylogarithmic is poorly
understood. We provide new and improved lower bounds. In a certain portion of
the parameter regime, those match the known upper bounds.
  Compared to the best previous lower bounds by Hao and Heydenreich, our result
has several advantages: it gives matching bounds for a larger range of
parameters, thus settling the question for a larger portion of the parameter
space. It strictly improves the lower bounds by Hao and Heydenreich for all
parameters settings in which those bounds were not tight. It gives tail bounds
on the probability of having short paths, which imply shape theorems for the
$k$-neighbourhood of a vertex whenever our lower bounds are tight, and tight
bounds for the size of this $k$-neighbourhood. And last but not least, our
proof is much simpler and not much longer than two pages, and we demonstrate
that it generalizes well by showing that the same technique also works for
first passage percolation.
