  Intermediate dimensions were recently introduced to provide a spectrum of
dimensions interpolating between Hausdorff and box-counting dimensions for
fractals where these differ. In particular, the self-affine Bedford-McMullen
carpets are a natural case for investigation, but until now only very rough
bounds for their intermediate dimensions have been found. In this paper, we
determine a precise formula for the intermediate dimensions $\dim_{\,
\theta}\Lambda$ of any Bedford-McMullen carpet $\Lambda$ for the whole spectrum
of $\theta \in [0,1]$, in terms of a certain large deviations rate function.
The intermediate dimensions exist and are strictly increasing in $\theta$, and
the function $\theta\mapsto \dim_{\, \theta}\Lambda$ exhibits interesting
features not witnessed on any previous example, such as having countably many
phase transitions, between which it is analytic and strictly concave.
  We make an unexpected connection to multifractal analysis by showing that two
carpets with non-uniform vertical fibres have equal intermediate dimensions if
and only if the Hausdorff multifractal spectra of the uniform Bernoulli
measures on the two carpets are equal. Since intermediate dimensions are
bi-Lipschitz invariant, this shows that the equality of these multifractal
spectra is a necessary condition for two such carpets to be Lipschitz
equivalent.
