  We show that localization in quasiperiodically modulated, two-dimensional
systems is stable to the presence of a finite density of ergodic grains. This
contrasts with the case of randomly modulated systems, where such grains seed
thermalizing avalanches. These results are obtained within a quantitatively
accurate, self-consistent entanglement mean field theory which analytically
describes two level systems connected to a central ergodic grain. The theory
predicts the distribution of entanglement entropies of each two level system
across eigenstates, and the late time values of dynamical observables. In
addition to recovering the known phenomenology of avalanches, the theory
reproduces exact diagonalization data, and predicts the spatial profile of the
thermalized region when the avalanche fails.
