  In this work, we study the information scrambling and the entanglement
dynamics in the complex Brownian Sachdev-Ye-Kitaev (cBSYK) models, focusing on
their dependence on the charge density $n$. We first derive the effective
theory for scramblons in a single cBSYK model, which gives closed-form
expressions for the late-time OTOC and operator size. In particular, the result
for OTOC is consistent with numerical observations in [1]. We then study the
entanglement dynamics in cBSYK chains. We derive the density dependence of the
entanglement velocity for both R\'enyi entropies and the Von Neumann entropy,
with a comparison to the butterfly velocity. We further consider adding
repeated measurements and derive the effective theory of the measurement
induced transition which shows $U(2)_L\otimes U(2)_R$ symmetry for
non-interacting models.
