Dynamics of operator size distribution in q-local quantum Brownian SYK and spin models

We study operator dynamics in Brownian quantum many-body models with q-local interactions. The operator dynamics are characterized by the time-dependent size distribution, for which we derive an exact master equation in both the Brownian Majorana Sachdev-Ye-Kitaev (SYK) model and the spin model for general q. This equation can be solved numerically for large systems. Additionally, we obtain the analytical size distribution in the large N limit for arbitrary initial conditions and q. The distributions for both models take the same form, related to the χ-squared distribution by a change of variable, and strongly depend on the initial condition. For small initial sizes, the operator dynamics are characterized by a broad distribution that narrows as the initial size increases. When the initial operator size is below q-2 for the Majorana model or q-1 for the spin model, the distribution diverges in the small size limit at all times. The mean size of all operators, which can be directly measured by the out-of-time ordered correlator, grows exponentially during the early time. In the late time regime, the mean size for a single Majorana or Pauli operator for all q decays exponentially as t e^-t, much slower than all other operators, which decay as e^-t. At finite N, the size distribution exhibits modulo-dependent branching within a symmetry sector for the q geq 8 Majorana model and the q geq 4 spin model. Our results reveal universal features of operator dynamics in q-local quantum many-body systems.