We consider the problem of evaluating the operator norm of the sum of operator-valued random Haar unitaries. We give compare it with the same sum where random Haar unitaries are replaced by free Haar unitaries and give criteria for the two norms to be close with high probability. We will describe the probabilistic and oprator theoretic tools needed to achieve such estimates. Time allowing, we will consider application to variants of the Peterson Thom problem. This talk is based on joint work with Charles Bordenave
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