We prove the tightness of a natural approximation scheme for an analog of the
Liouville quantum gravity metric on Rd for arbitrary d≥2. More
precisely, let {hn​}n≥1​ be a suitable sequence of Gaussian random
functions which approximates a log-correlated Gaussian field on Rd.
Consider the family of random metrics on Rd obtained by weighting
the lengths of paths by eξhn​, where ξ>0 is a parameter. We prove
that if ξ belongs to the subcritical phase (which is defined by the
condition that the distance exponent Q(ξ) is greater than 2d​),
then after appropriate re-scaling, these metrics are tight and that every
subsequential limit is a metric on Rd which induces the Euclidean
topology. We include a substantial list of open problems.Comment: 68 pages, 8 figures; version 2 has updated reference