  We present a prototype for Wilsonian analysis of asymptotics of
supersymmetric partition functions of non-abelian gauge theories. Localization
allows expressing such partition functions as an integral over a BPS moduli
space. When the limit of interest introduces a scale hierarchy in the problem,
asymptotics of the partition function is obtained in the Wilsonian approach by
$i)$ decomposing (in some suitable scheme) the BPS moduli space into various
patches according to the set of light fields (lighter than the scheme dependent
cut-off $\Lambda$) they support, $ii)$ localizing the partition function of the
effective field theory on each patch (with cut-offs set by the scheme), and
$iii)$ summing up the contributions of all patches to obtain the final
asymptotic result (which is scheme-independent and accurate as
$\Lambda\to\infty$). Our prototype concerns the Cardy-like asymptotics of the
4d superconformal index, which has been of interest recently for its
application to black hole microstate counting in AdS$_5$/CFT$_4$. As a
byproduct of our analysis we obtain the most general asymptotic expression for
the index of gauge theories in the Cardy-like limit, encompassing and extending
all previous results.
