  Two of the dominant channels to produce the black-hole binary mergers
observed by LIGO and Virgo are believed to be the isolated evolution of stellar
binaries in the field and dynamical formation in star clusters. Their relative
efficiency can be characterized by a "mixing fraction." Pair instabilities
prevent stellar collapse from generating black holes more massive than about
$45 M_\odot$. This "mass gap" only applies to the field formation scenario, and
it can be filled by repeated mergers in clusters. A similar reasoning applies
to the binary's effective spin. If black holes are born slowly rotating, the
high-spin portion of the parameter space (the "spin gap") can only be populated
by black hole binaries that were assembled dynamically. Using a semianalytical
cluster model, we show that future gravitational-wave events in either the mass
gap, the spin gap, or both can be leveraged to infer the mixing fraction
between the field and cluster formation channels.
