When the fundamental parameters of a theory are functions of a scalar field subject to large fluctuations during inflation, quantum phase transitions can act as dynamical attractors. As a result, the theory parameters can be probabilistically localised around critical values, with the Universe finding itself at the edge of a phase transition. We illustrate how this mechanism could account for the observed near-criticality of the Higgs vacuum, as well as an explanation for the naturalness of the Higgs mass.