  Many stellar-mass Black Holes (sBHs) are expected to orbit supermassive black
holes at galactic centers. For galaxies with Active Galactic Nuclei (AGN), it
is likely that the sBHs reside in a disk. We study the formation of sBH
binaries via gravitational wave emission in such disks. We examine analytically
the dynamics of two sBHs orbiting a supermassive black hole, estimate the
capture cross section, and derive the eccentricity distribution of bound
binaries at different frequency bands. We find that the majority of the merging
sBH binaries, assembled in this manner, can be measured as highly eccentric,
detectable in the LIGO-Virgo-KAGRA (LVK) band from their formation, with
$(1-e)\ll1$, through their circularization and up to their merger; the
remaining binaries circularize to small eccentricities ($e\lesssim0.3$) before
entering the LVK band. More eccentric mergers would be observed for sBHs with
higher random velocities, closer to the supermassive black hole, or at lower
observing frequency bands, as planned in future gravitational wave detectors
such as the Einstein Telescope and LISA.
