Strong convergence of tensor products of independent G.U.E. matrices

Abstract

Given tuples of properly normalized independent $N\times N$ G.U.E. matrices $(X_N^{(1)},\dots ,X_N^{(r_1)})$ and $(Y_N^{(1)},\dots ,Y_N^{(r_2)})$, we show that the tuple $(X_N^{(1)}\otimes I_N,\dots ,X_N^{(r_1)}\otimes I_N,I_N\otimes Y_N^{(1)},\dots ,I_N\otimes Y_N^{(r_2)})$ of $N^2\times N^2$ random matrices converges strongly as $N$ tends to infinity. It was shown by Ben Hayes that this result implies that the Peterson-Thom conjecture is true.

Second, longer version, providing explicit calculations for the application of the flip and the partial difference-differential on resolvents of tensor products of operators. The reader comfortable with free noncommutative functions theory might benefit more from reading the first version

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