Abstract
We introduce the concept of matrix liberation process, a random matrix counterpart of the liberation process in free probability, and prove a large deviation upper bound for its empirical distribution and several properties on its rate function. As a simple consequence, we obtain the almost sure convergence of the empirical distribution of the matrix liberation process to that of the corresponding liberation process as continuous processes in the large N limit.
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Acknowledgements
We would like to express our sincere gratitude to the referee for his/her very careful reading of this paper and pointing out a mistake in the original proof of exponential tightness.
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This work was supported by Japan Society for the Promotion of Science (Grant-in-Aid for Challenging Exploratory Research JP16K13762).
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Ueda, Y. Matrix Liberation Process I: Large Deviation Upper Bound and Almost Sure Convergence. J Theor Probab 32, 806–847 (2019). https://doi.org/10.1007/s10959-018-0819-z
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DOI: https://doi.org/10.1007/s10959-018-0819-z