Navier Stokes hydrodynamics in a random quantum circuit

Typical many body lattice quantum systems thermalize with the approach being governed by a universal effective description in terms of transport of conserved quantities known as hydrodynamics. While one dimensional lattice quantum systems are known to give rise to a variety of hydrodynamics, much less is known about the hydrodynamics that emerges in two dimensional lattice quantum systems. In this work, we construct a two dimensional random quantum circuit whose averaged dynamics is described by a variant of the FHP model, a well known classical Markov process governed by Navier-Stokes hydrodynamics at late times. Furthermore, we describe how circuit to circuit fluctuations of transport observables can be mapped onto an effective Markov process corresponding to coupled copies of the original modified FHP model. We numerically explore whether this coupling is irrelevant which corresponds to a typical circuit obeying the modified FHP model at late times.