Tightness of exponential metrics for log-correlated Gaussian fields in arbitrary dimension

We prove the tightness of a natural approximation scheme for an analog of the Liouville quantum gravity metric on ${\mathbb{R}}^d$ for arbitrary $d\ge 2$. More precisely, let $\{h_n{\}}_{n\ge 1}$ 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.