The article explores a synchronous particle movement process on a complete graph, focusing on the dispersion time. It analyzes the critical window where particles' number corresponds to the graph's vertices. The dispersion time is studied in relation to the critical window, showing convergence to a logistic branching process. Explicit asymptotics are formulated for transitions in and out of the critical window. The total number of jumps is also analyzed, converging to a fixed quantity when scaled by n ln n. The study provides insights into the abrupt transition from logarithmic to exponential time in dispersion processes.
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