Large Deviations of the Schwarzian Field Theory

We prove a large deviations principle for the probabilistic Schwarzian Field Theory at low temperatures. We demonstrate that the good rate function is equal to the action of the Schwarzian Field Theory, and we find its minimisers. In addition, we define an analogue of the H\"{o}lder condition on the functional space $\mathrm{Diff}^1(\mathbb{T})/\mathrm{PSL}(2,\mathbb{R})$ in terms of cross-ratio observables, characterise them in terms of the usual H\"{o}lder property on the space of continuous functions, and deduce the corresponding compact embedding theorem. We also show that the Schwarzian measure concentrates on functions satisfying the defined condition.