  We consider random walks on finite vertex-transitive graphs $\Gamma$ of
bounded degree. We find a simple geometric condition which characterises the
cover time fluctuations: the suitably normalised cover time converges to a
standard Gumbel variable if and only if $\mathrm{Diam}(\Gamma)^2 = o(n/\log
n)$, where $n = |\Gamma|$. We prove that this condition is furthermore
equivalent to the decorrelation of the uncovered set. The arguments rely on
recent breakthroughs by Tessera and Tointon on finitary versions of Gromov's
theorem on groups of polynomial growth, which we leverage into strong heat
kernel bounds, and refined quantitative estimates on Aldous and Brown's
exponential approximation of hitting times, which are of independent interest.
