This paper (link) uses a cosmological hydro simulation to explore the Edd. Ratios (ER) of the black holes that they seed in their simulation. A BH of mass 10^5 SM is seeded in haloes over 10^10 SM. They allow for Bondi-Hoyle Accretion with a modified parameter for the gas density close to the BH and compare it with Edd. Accretion to get the ERs. Various feedback effects are accounted for and they also plot the ER carefully so as to avoid any inherent trends. They find that the ER depends mostly on the the density of the gas being accreted.
They fit a universal log-normal distribution to the Edd. Ratio and compare it with observations. The Edd. Ratios they find are mostly < 1 implying sub-Eddnington accretion for the massive BH seeds in their work.
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