The "visible" Wigner matrix

Arup Bose; Soumendu Sundar Mukherjee

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arXiv · 2312.12428

The "visible" Wigner matrix

Arup Bose·Soumendu Sundar Mukherjee

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Abstract

We consider the ``visible'' Wigner matrix, a Wigner matrix whose $(i, j)$-th entry is coerced to zero if $i, j$ are co-prime. Using a recent result from elementary number theory on co-primality patterns in integers, we show that the limiting spectral distribution of this matrix exists, and give explicit descriptions of its moments in terms of infinite products over primes $p$ of certain polynomials evaluated at $1/p$. We also consider the complementary ``invisible'' Wigner matrix.

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Arup Bose, Soumendu Sundar Mukherjee. 2023-12-19. The "visible" Wigner matrix. https://arxiv.org/abs/2312.12428

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