Black hole evolution in semiclassical gravity has been investigated in 1+1 dimensional models which serve as toy models for spherical symmetry (see  for a review). The black hole threshold in such models has been investigated in [45, 9, 194, 137, 210, 174, 31]. In some of these models, the critical exponent is 1/2 for kinematical reasons.
In  a 3+1-dimensional but perturbative approach is taken. The quantum effects then give an additional unstable mode with = 2. If this is larger than the positive Lyapunov exponent , it will become the dominant perturbation for sufficiently good fine-tuning, and therefore sufficiently good fine-tuning will reveal a mass gap. The mass gap is found also in numerical evolutions of a spherical scalar field in 3+1 dimensions with the semiclassical equations obtained in the framework of “singularity resolution” in loop quantum gravity [132, 211]. Update
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