$T\bar{T}$ deformation of random matrices

We define and study the $T\bar{T}$ deformation of a random matrix model, showing a consistent definition requires the inclusion of both the perturbative and non-perturbative solutions to the flow equation. The deformed model is well defined for arbitrary values of the coupling, exhibiting a phase transition for the critical value in which the spectrum complexifies. The transition is between a single and a double-cut phase, typically third order and in the same universality class as the Gross-Witten transition in lattice gauge theory. The $T\bar{T}$ deformation of a double scaled model is more subtle and complicated, and we are not able to give a compelling definition, although we discuss obstacles and possible alternatives. Preliminary comparisons with finite cut-off Jackiw-Teitelboim gravity are presented.