Ewan McCulloch-UMass: Full counting statistics of charge in chaotic many-body quantum systems

We investigate the full counting statistics of charge transport in U(1)-symmetric random unitary circuits. We consider an initial mixed state prepared with a chemical potential imbalance between the left and right halves of the system and study the fluctuations of the charge transferred across the central bond in typical circuits. Using an effective replica statistical mechanics model and a mapping onto an emergent classical stochastic process valid at large onsite Hilbert space dimension, we show that charge transfer fluctuations approach those of the symmetric exclusion process at long times, with sub-leading quantum corrections. Our results establish the emergence of ``classicality'' at long times in quantum systems, even at the level of fluctuations.