  Asymptotic safety is a powerful mechanism for obtaining a consistent and
predictive quantum field theory beyond the realm of perturbation theory. It
hinges on an interacting fixed point of the Wilsonian renormalization group
flow which controls the microscopic dynamics. Connecting the fixed point to
observations requires constructing the set of effective actions compatible with
this microscopic dynamics. Technically, this information is stored in the
UV-critical surface of the fixed point. In this work, we describe a novel
approach for extracting this information based on analytical and
pseudo-spectral methods. Our construction is illustrated at the level of the
two-dimensional Ising model and easily generalizes to any asymptotically safe
quantum field theory. It also constitutes an important step towards setting up
a well-founded swampland program within the gravitational asymptotic safety
program.
