The most massive black holes observed in the Universe weigh up to \sim 10 ^ { 10 } \mathrm { M _ { \odot } } , nearly independent of redshift .
Reaching these final masses likely required copious accretion and several major mergers .
Employing a dynamical approach , that rests on the role played by a new , relevant physical scale - the transition radius - we provide a theoretical calculation of the maximum mass achievable by a black hole seed that forms in an isolated halo , one that scarcely merged .
Incorporating effects at the transition radius and their impact on the evolution of accretion in isolated haloes we are able to obtain new limits for permitted growth .
We find that large black hole seeds ( M _ { \bullet } \mathrel { \hbox to 0.0 pt { \lower 4.0 pt \hbox { $ \sim$ } } \raise 1.0 pt \hbox% { $ > $ } } 10 ^ { 4 } \mathrm { M _ { \odot } } ) hosted in small isolated halos ( M _ { h } \lesssim 10 ^ { 9 } \mathrm { M _ { \odot } } ) accreting with relatively small radiative efficiencies ( \epsilon \lesssim 0.1 ) grow optimally in these circumstances .
Moreover , we show that the standard M _ { \bullet } - \sigma relation observed at z \sim 0 can not be established in isolated halos at high- z , but requires the occurrence of mergers .
Since the average limiting mass of black holes formed at z \mathrel { \hbox to 0.0 pt { \lower 4.0 pt \hbox { $ \sim$ } } \raise 1.0 pt \hbox { $ > $ } } 10 is in the range 10 ^ { 4 - 6 } \mathrm { M _ { \odot } } , we expect to observe them in local galaxies as intermediate-mass black holes , when hosted in the rare haloes that experienced only minor or no merging events .
Such ancient black holes , formed in isolation with subsequent scant growth , could survive , almost unchanged , until present .