  Intuitively, two metric spaces are rough isometric (or quasi-isometric) if
their large-scale metric structure is the same, ignoring fine details. This
concept has proven fundamental in the geometric study of groups. Ab\'{e}rt, and
later Szegedy and Benjamini, have posed several probabilistic questions
concerning this concept. In this article, we consider one of the simplest of
these: are two independent Poisson point processes on the line rough isometric
almost surely? Szegedy conjectured that the answer is positive. Benjamini
proposed to consider a quantitative version which roughly states the following:
given two independent percolations on $\mathbb {N}$, for which constants are
the first $n$ points of the first percolation rough isometric to an initial
segment of the second, with the first point mapping to the first point and with
probability uniformly bounded from below? We prove that the original question
is equivalent to proving that absolute constants are possible in this
quantitative version. We then make some progress toward the conjecture by
showing that constants of order $\sqrt{\log n}$ suffice in the quantitative
version. This is the first result to improve upon the trivial construction
which has constants of order $\log n$. Furthermore, the rough isometry we
construct is (weakly) monotone and we include a discussion of monotone rough
isometries, their properties and an interesting lattice structure inherent in
them.
