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PreprintarXiv licencearXiv

Least singular values of Gaussian matrices with deterministic shifts

  • Xuanang Hu
  • Zeyan Song
  • Xinglong Wu
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Abstract

Let A be a deterministic real square matrix and G a standard Gaussian matrix. We determine Esmin(A+G), up to universal constants, in terms of the singular values of A. The expectation is comparable to the square root of the smallest zero of the expected characteristic polynomial of (A+G)⊤(A+G). This resolves Problem~3 in Tikhomirov's ICM 2022 survey. We also prove that the zeros of the expected characteristic polynomial are majorized by the expected ordered eigenvalues of (A+G)⊤(A+G).

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Cite

BibTeXRIS
BibTeX
@misc{hu2026least,
  title = {{Least singular values of Gaussian matrices with deterministic shifts}},
  author = {Xuanang Hu and Zeyan Song and Xinglong Wu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.27008},
  eprint = {2610.27008},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.27008},
}
RIS
TY  - UNPB
TI  - Least singular values of Gaussian matrices with deterministic shifts
AU  - Xuanang Hu
AU  - Zeyan Song
AU  - Xinglong Wu
PY  - 2026
DA  - 2026/10/19
PB  - arXiv
DO  - 10.48550/arxiv.2610.27008
UR  - https://arxiv.org/abs/2610.27008
AB  - Let A be a deterministic real square matrix and G a standard Gaussian matrix. We determine Esmin(A+G), up to universal constants, in terms of the singular values of A. The expectation is comparable to the square root of the smallest zero of the expected characteristic polynomial of (A+G)^⊤(A+G). This resolves Problem~3 in Tikhomirov's ICM 2022 survey. We also prove that the zeros of the expected characteristic polynomial are majorized by the expected ordered eigenvalues of (A+G)^⊤(A+G).
ER  -