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We introduce the concept of matrix liberation process, a random matrixcounterpart of the liberation process in free probability, and prove a largedeviation upper bound for its empirical distribution with several properties onits rate function. As a simple consequence we obtain the almost sureconvergence of the empirical distribution of the matrix liberation process tothat of the corresponding liberation process as continuous processes in large$N$ limit.

Matrix liberation process I: Large deviation upper bound and almost sure convergence