EMRI eccentricity probability density function at the mostly bound orbit. The blue line corresponds to the case where the sBH number fraction is larger than the critical value, ﹩{{\rm{f}}}_{\bullet }\geqslant {f}_{\bullet }^{c}﹩, where the distribution is independent of the specific value of f _•, and peaks at e ≃ 0.08. The red line exhibits an example of the complementary case, ﹩{{\rm{f}}}_{\bullet }={10}^{-5}\lt {f}_{\bullet }^{c}﹩, where the distribution peaks at e _I ≃ 0.04 (as given by Equation (29)) and falls steeply, as ﹩p(e\gtrsim {e}_{I})\mathop{\propto }\limits_{\sim }{e}^{-49/19}﹩. In both cases, the distributions satisfy p(e ≪ 1) ∝ e ^5/19.