Spectrum of the steady-state distribution function of disordered Lindbladian fermions at large channel number

Free fermions coupled to particle gain and loss relax to a Gaussian steady state, in which every natural orbital has its own occupation. We find the distribution of these occupations $\varrho_{\rm st}$ for a random Hamiltonian coupled to many random gain and loss channels. Each orbital is filled and emptied at its own rates, and the Hamiltonian, moving particles across the levels during their lifetime, averages the differences between the orbitals. We show that this averaging is incomplete: the spread of the occupations decreases only as the inverse square root of the number of channels per level. At large system size the distribution depends on the grain Hamiltonian only through its density of states $ν(E)$. The spread is bounded by the baths alone: even Hamiltonians with unbounded, heavy-tailed spectra cannot widen it beyond this bound. The spread suppresses the particle-number fluctuations below their binomial value.