Two metric spaces are said to be quasi isometric if their metrics are equivalent up to multiplicative and additive constants. This notion, introduced by Gromov (1981) for
groups and more generally by Kanai (1985) has proved to be important in coarse geometry. There has been recent interest in understanding the coarse geometry of random subgraphs of Cayley graphs and in particular whether or not two independent copies of random metric spaces with identical distribution are quasi isometric. I shall describe a sequence of works (joint with Allan Sly and Vladas Sidoravicius) that address questions of this flavour for random subsets of Euclidean Spaces. In particular we show that two copies of Bernoulli percolation on $\mathbb{Z}$; i.e., when each element of $\mathbb{Z}$ is included in the random subset independently and with equal probability, are almost surely quasi isometric. We develop a multi-scale framework flexible enough to tackle a number of such embedding questions including some closely related natural questions in dependent percolation. I shall also discuss an approach to higher dimensional extensions and a number of interesting open questions.
