  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.
