Large deviations for the interchange process on the interval and incompressible flows

In the talk I will explain how large deviations for the one-dimensional interchange process are related to incompressible Euler equations. I will sketch how to obtain matching large deviation lower and upper bounds for processes which are energy-minimizing (under certain additional regularity assumptions), putting the work in the context of generalized incompressible flows introduced by Brenier. Joint work with Bálint Virág.