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PreprintCC BY 4.0arXiv

Spectral properties of Random Matrices

  • Anastasis Kafetzopoulos
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Abstract

We give the theoretical foundations of random matrix theory through the definitions of a random matrix, a random probability measure and the corresponding empirical spectral distribution. The technical tool we use is the Stieltjes transform method through which we prove optimal convergence of the empirical spectral distribution of random sample covariance matrices to the deterministic Marchenko-Pastur distribution. We also give new results about the rigidity of the eigenvalues of this random sample covariance matrix and the rate of their convergence. We then define the Dyson equation method to prove new local laws about a random matrix model that interpolates between the Marchenko-Pastur distribution, the elliptical law and the circular law. Through our work these local laws can be considered universal.

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Cite

BibTeXRIS
BibTeX
@misc{kafetzopoulos2026spectral,
  title = {{Spectral properties of Random Matrices}},
  author = {Anastasis Kafetzopoulos},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.11011},
  eprint = {2609.11011},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.11011},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Spectral properties of Random Matrices
AU  - Anastasis Kafetzopoulos
PY  - 2026
DA  - 2026/09/10
PB  - arXiv
DO  - 10.48550/arxiv.2609.11011
UR  - https://arxiv.org/abs/2609.11011
AB  - We give the theoretical foundations of random matrix theory through the definitions of a random matrix, a random probability measure and the corresponding empirical spectral distribution. The technical tool we use is the Stieltjes transform method through which we prove optimal convergence of the empirical spectral distribution of random sample covariance matrices to the deterministic Marchenko-Pastur distribution. We also give new results about the rigidity of the eigenvalues of this random sample covariance matrix and the rate of their convergence. We then define the Dyson equation method to prove new local laws about a random matrix model that interpolates between the Marchenko-Pastur distribution, the elliptical law and the circular law. Through our work these local laws can be considered universal.
ER  -