This paper estimates the specific accretion-rate distribution of AGN using a sample of 4821 X-ray sources from both deep and shallow surveys. The specific accretion-rate distribution is defined as the probability of a galaxy with a given stellar mass and redshift hosting an active nucleus with a certain specific accretion rate. We find that the probability of a galaxy hosting an AGN increases with decreasing specific accretion rate. There is evidence that this trend reverses at low specific accretion rates, $lambda<10^{-4}-10^{-3}$ (in Eddington units). There is also a break close to the Eddington limit, above which the probability of an accretion event decreases steeply. The specific accretion-rate distribution evolves such that the fraction of AGN among galaxies drops toward lower redshifts. This decrease in the AGN duty cycle is responsible for the strong evolution of the accretion density of the Universe from redshift $zapprox1-1.5$ to the present day. Our analysis also suggests that this evolution is accompanied by a decoupling of accretion events onto black holes from the formation of stars in galaxies. There is also evidence that at earlier times the relative probability of high vs low specific accretion-rate events among galaxies increases. We argue that this differential redshift evolution of the AGN duty cycle with respect to $lambda$ produces the AGN downsizing trend, whereby luminous sources peak at earlier epochs compared to less luminous ones. Finally, we also find a stellar-mass dependence of the specific accretion-rate distribution, with more massive galaxies avoiding high specific accretion-rate events.
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