We present Trinity, a flexible empirical model that self-consistently infers the statistical connection between dark matter haloes, galaxies, and supermassive black holes (SMBHs). Trinity is constrained by galaxy observables from $0 < z < 10$ (galaxies stellar mass functions, specific and cosmic SFRs, quenched fractions, and UV luminosity functions) and SMBH observables from $0 < z < 6.5$ (quasar luminosity functions, quasar probability distribution functions, active black hole mass functions, local SMBH mass-bulge mass relations, and the observed SMBH mass distributions of high redshift bright quasars). The model includes full treatment of observational systematics (e.g., AGN obscuration and errors in stellar masses). From these data, Trinity infers the average SMBH mass, SMBH accretion rate, merger rate, and Eddington ratio distribution as functions of halo mass, galaxy stellar mass, and redshift. Key findings include: 1) the normalization of the SMBH mass-bulge mass relation increases only mildly from $z=0$ to $z=3$, but decreases more strongly from $z=3$ to $z=10$; 2) The AGN radiative$+$kinetic efficiency is $sim$0.04-0.07, and does not show significant redshift dependence given the existing data constraints; 3) AGNs show downsizing, i.e., the Eddington ratios of more massive SMBHs start to decrease earlier than those of lower-mass objects; 4) The average ratio between average SMBH accretion rate and SFR is $sim10^{-3}$ for low-mass galaxies, which are primarily star-forming. This ratio increases to $sim10^{-1}$ for the most massive haloes below $zsim1$, where star formation is quenched but SMBHs continue to accrete.
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