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$T\overline{T}$-deformed free energy of the Airy model

Sharpening the holographic correspondence of Jackiw-Teitelboim (JT) gravity and its dual matrix model description at a finite radial cutoff $λ$ through the $T\overline{T}$ deformation is of interest. To proceed, we simplify the problem by considering the Airy model and deform Airy correlators in the same way as in $T\overline{T}$-deformed JT gravity. We use those correlators to compute the annealed and quenched free energies for both $λ> 0$ and $λ< 0$ from an integral representation of the replica trick. At the leading order in $λ$ and low temperatures, we confirm that the genus-zero quenched free energy monotonically decreases as a function of temperature when perturbation theory is valid. We then study the all-genus quenched free energy at low temperatures, where we discover and discuss subtleties due to non-perturbative effects in the Airy model, as well as the contributions from the non-perturbative branch under the $T\overline{T}$ deformation.