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Random symmetric matrices: rank distribution and irreducibility of the characteristic polynomial

Part of: Special matrices Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  12 May 2022

ASAF FERBER ,
VISHESH JAIN ,
ASHWIN SAH  and
MEHTAAB SAWHNEY
ASAF FERBER
Affiliation:
340 Rowland Hall (Building 400), University of California, Irvine, Irvine CA 92697, U.S.A. e-mail: asaff@uci.edu
VISHESH JAIN
Affiliation:
390 Jane Stanford Way, Stanford, CA 94305, U.S.A. e-mail: visheshj@stanford.edu
ASHWIN SAH
Affiliation:
Simons Building (Building 2), 77 Massachusetts Avenue, Cambridge MA 02139, U.S.A. e-mails: asah@mit.edu, msawhney@mit.edu
MEHTAAB SAWHNEY
Affiliation:
Simons Building (Building 2), 77 Massachusetts Avenue, Cambridge MA 02139, U.S.A. e-mails: asah@mit.edu, msawhney@mit.edu
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Abstract

Conditional on the extended Riemann hypothesis, we show that with high probability, the characteristic polynomial of a random symmetric $\{\pm 1\}$ -matrix is irreducible. This addresses a question raised by Eberhard in recent work. The main innovation in our work is establishing sharp estimates regarding the rank distribution of symmetric random $\{\pm 1\}$ -matrices over $\mathbb{F}_p$ for primes $2 < p \leq \exp(O(n^{1/4}))$ . Previously, such estimates were available only for $p = o(n^{1/8})$ . At the heart of our proof is a way to combine multiple inverse Littlewood–Offord-type results to control the contribution to singularity-type events of vectors in $\mathbb{F}_p^{n}$ with anticoncentration at least $1/p + \Omega(1/p^2)$ . Previously, inverse Littlewood–Offord-type results only allowed control over vectors with anticoncentration at least $C/p$ for some large constant $C > 1$ .

MSC classification

Primary: 15B52: Random matrices
Secondary: 11M50: Relations with random matrices
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 174 , Issue 2 , March 2023 , pp. 233 - 246
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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Footnotes

†

Supported in part by NSF grants DMS-1954395 and DMS-1953799.

‡

Sah and Sawhney were supported by NSF Graduate Research Fellowship Program DGE-1745302.

References

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