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.
- Publication:
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arXiv e-prints
- Pub Date:
- May 2022
- DOI:
- 10.48550/arXiv.2205.07695
- arXiv:
- arXiv:2205.07695
- Bibcode:
- 2022arXiv220507695B
- Keywords:
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- Mathematics - Operator Algebras;
- Mathematics - Probability
- E-Print:
- 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