Perturbations of an exact solution for (2+1)-dimensional critical collapse

We find the perturbation spectrum of a family of spherically symmetric and continuously self-similar (CSS) exact solutions that appear to be relevant for the critical collapse of scalar field matter in $2+1$ spacetime dimensions. The rate of exponential growth of the unstable perturbation yields the critical exponent. Our results are compared to the numerical simulations of Pretorius and Choptuik and are inconclusive: We find a CSS solution with exactly one unstable mode, which suggests that it may be the critical solution, but another CSS solution which has three unstable modes fits the numerically found critical solution better.