Note on entropy dynamics in the Brownian SYK model

We study the time evolution of R\'enyi entropy in a system two coupled Brownian SYK clusters evolving from an initial product state. The one cluster grows linearly and then saturates to coarse grained entropy. This Page curve is obtained by different methods, path integral saddle point analysis operator dynamics analysis. Using character dynamics, we derive master equation which controls gives for purity. Insight into physics this complicated provided complementary method: replica diagonal non-diagonal saddles are responsible linear growth saturation entropy, respectively.