Do we expect most AGN to live in discs?

Recent observations have indicated that a large fraction of the low-to intermediate-luminosity AGN population lives in disc-dominated hosts, while the more luminous quasars live in bulge-dominated hosts (that may or may not be major merger remnants), in conflict with some previous model predictions. We therefore build and compare a semi-empirical model for AGN fuelling which accounts for both merger and non-merger 'triggering'. In particular, we show that the 'stochastic accretion' model - in which fuelling in disc galaxies is essentially a random process arising whenever dense gas clouds reach the nucleus - provides a good match to the present observations at low/intermediate luminosities. However, it falls short of the high-luminosity population. We combine this with models for major merger-induced AGN fueling, which lead to rarer but more luminous events, and predict the resulting abundance of disc-dominated and bulge-dominated AGN host galaxies as a function of luminosity and redshift. We compile and compare observational constraints from z similar to 0 to 2. The models and observations generically show a transition from disc to bulge dominance in hosts near the Seyfert-quasar transition, at all redshifts. 'Stochastic' fuelling dominates AGN by number (dominant at low luminosity), and dominates black hole (BH) growth below the 'knee' in the present-day BH mass function (less than or similar to 10(7)M(circle dot)). However, it accounts for just similar to 10 per cent of BH mass growth at masses greater than or similar to 10(8)M(circle dot). In total, fuelling in discy hosts accounts for similar to 30 per cent of the total AGN luminosity density/BH mass density. The combined model also accurately predicts the AGN luminosity function and clustering/bias as a function of luminosity and redshift; however, we argue that these are not sensitive probes of BH fuelling mechanisms.