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 binarys 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.
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