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A tensor, or hypermatrix, is a higher order generalization of a matrix. Just as a matrix can be multiplied by a vector to produce another vector, a tensor can be contracted to produce a lower order object. For a random Wigner-type tensor, we study the family of random matrices obtained by unit vector contractions. In general, the entries of such a matrix are dependent. Nevertheless, we show that the joint spectral distribution of this ensemble is well-approximated by a semicircular family with an explicit covariance structure given by the rescaled overlaps of the symmetric tensor product of the contracting vectors. In the single-matrix case, this implies convergence of the empirical spectral distribution to a semicircle distribution whose variance depends on the alignment of the contracting vectors. Here, we find a connection to a combinatorial object known as a uniform block permutation. Our analysis relies on a tensorial extension of the usual graphical calculus for moment method calculations in random matrix theory. This is joint work with Jorge Garza-Vargas (UC Berkeley).

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