  We prove the tightness of a natural approximation scheme for an analog of the
Liouville quantum gravity metric on $\mathbb R^d$ for arbitrary $d\geq 2$. More
precisely, let $\{h_n\}_{n\geq 1}$ be a suitable sequence of Gaussian random
functions which approximates a log-correlated Gaussian field on $\mathbb R^d$.
Consider the family of random metrics on $\mathbb R^d$ obtained by weighting
the lengths of paths by $e^{\xi h_n}$, where $\xi > 0$ is a parameter. We prove
that if $\xi$ belongs to the subcritical phase (which is defined by the
condition that the distance exponent $Q(\xi)$ is greater than $\sqrt{2d}$),
then after appropriate re-scaling, these metrics are tight and that every
subsequential limit is a metric on $\mathbb R^d$ which induces the Euclidean
topology. We include a substantial list of open problems.
