Large Deviations of the Schwarzian Field Theory

TL;DR

This paper establishes a large deviations principle for the Schwarzian Field Theory at low temperatures, linking the rate function to the theory's action and analyzing the measure's concentration properties.

Contribution

It proves a large deviations principle for the Schwarzian Field Theory, identifies the rate function with its action, and introduces a new H"{o}lder condition for the functional space.

Findings

Rate function equals the Schwarzian action.

Concentration of measure on functions satisfying the H"{o}lder-like condition.

Characterization of the condition via cross-ratio observables.

Abstract

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.