### 4.6 Quantum effects

Type II critical phenomena provide a relatively natural way of producing arbitrarily high curvatures,
where quantum gravity effects should become important, from generic initial data. Approaching the Planck
scale from above, one would expect to be able to write down a critical solution that is the classical critical
solution asymptotically at large scales, as an expansion in inverse powers of the Planck length (see
Section 2.6).
Black hole evolution in semiclassical gravity has been investigated in 1+1 dimensional models which
serve as toy models for spherical symmetry (see [92] 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 [36] 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