Let $X, Y$ be two random graphs on $n$ vertices (say, in $G(n, p)$ model for some $p$). Can anything (expectation, value with high probabiity, ...) be said about $D(X, Y)$, where $D$ is the minimal distortion of a metric embedding $X \rightarrow Y$?

Recall that for two metric spaces $(X, d), (Y, d')$ a Lipschitz map $f: X\rightarrow Y$ has distortion $C$ if $d(x,x')\leq C d'(f(x),f(x'))$.

In some sense this asks if the finite coarse geometry of a random graph is unique. Since the maximal distortion is always $O(\log n)$ (Bourgain's theorem), the essential question is whether $D(X, Y)=\Omega(\log n)$ or $D(X,Y)=o(\log n)$ with high probability. One can e.g. ask whether two sequences of random graphs $\{X_n\}_{n=1}^{\infty}, \{Y_n\}_{n=1}^{\infty}$ coarsely embed into each other.