How likely are quantum glitches during inflation to birth black holes?

During cosmic inflation, quantum noise can occasionally kick a field far enough to seed primordial black holes — but calculating exactly how likely that is has been technically painful. This work reframes the problem as an eigenvalue equation, recovering known results and exposing a previously overlooked power-law regime (P ∝ N^{−3/2}) sitting between the distribution's peak and its exponential tail. For the physically interesting 'broad-well' case mimicking features in the inflaton potential, the tail is dramatically suppressed compared to simpler models — directly affecting primordial black hole abundance predictions.