Determining the structure of the UV-critical hypersurface, spanned by renormalization group trajectories enjoying asymptotic safety, is a highly complex problem in the gravitational asymptotic safety program. In this talk, I will outline how the composite operator formalism for the effective average action can be used to estimate the spectrum of the stability matrix encoding the linearized RG flow at the Reuter fixed point. On this basis, we identify a perturbative regime where the spectral properties are governed by canonical power counting. Moreover, the explicit analysis of the eigenvalue spectrum up to N=100 reveals intriguing distributions of eigenvalues in the complex plane reminiscent of Lee-Yang theory. The talk is based on two recent works arXiv:2002.00256 and arXiv:2003.07454. An attempt will be made to include animations in the seminar.