ÍâÍøÌìÌÃ

Given $k$ iid $n\times n$ random Haar unitaries $U_1,\ldots , U_k$Ìýand $l$ a fixed integer, we consider the jointÌýbehavior of $U_1^{\otimesÌýl},\ldots , U_k^{\otimes l}$ and show that this sequence of $k$-tuples isÌýalmost surely strongly asymptotically free in the large $n$ limit. StrongÌýasymptotic freeness is a particular case of strong convergence, whichÌýensures the absence of outliers for a matrix model obtained from aÌýnon-commutative polynomial in the $k$-tupleÌý We will explain ourÌýmotivations, describe some variants of our result and some applications inÌýasymptotic representation theory. We will also elaborate on a few salientÌýaspects of the proof, such as a theory of matrix valued non-backtrackingÌýoperators, and a new inequality between moments of gaussian and unitaryÌýmatrices. This talk is based on joint work in preparation with CharlesÌýBordenave.