This note describes a small step in the analysis of the fully asynchronous cellular automata (i.e., the cells are updated uniformly at random at each time step). We establish the rapid convergence of fifteen minimal Elementary Cellular Automata, showing that their average convergence time to a fixed point scales logarithmically with the size of the automaton. Techniques involve the use of Markov chain analysis and the construction of adequate potential functions. The problem is however left open for twelve other minimal rules, which shows the need to develop this analysis further.