The anisotropic local law for sample covariance matrices under quadratic-form concentration
Abstract
We study sample covariance matrices K = 1/N ∑i=1N xi xi* ∈ ℝn × n in the proportional regime n asymp N. The columns $ \mathbf{x}_1, \ldots, \mathbf{x}_N \in \mathbb{R}^n$ are independent and centered, with common covariance 𝔼 xi xi* = Σ, but may otherwise have strongly and nonlinearly dependent coordinates. Assuming only that quadratic forms of the columns concentrate uniformly at the optimal rate | xi* A xi − Tr ΣA | ≺ | A |F, together with polynomial norm moments and a standard nondegeneracy condition on Σ, we prove the optimal anisotropic local law: on regular spectral domains, uniformly down to spectral scales η:= Im z ≥ N−1 + τ, | ⟨ u , ( (K−z)−1 − (−zIn−zm̃0(z)Σ)−1 ) v > | ≺ √((Im m̃0 (z))/Nη) + 1/Nη for all deterministic unit vectors $ \mathbf{u}, \mathbf{v} \in \mathbb{C}^n$, where m̃0(z) is the Stieltjes transform of the deformed Marchenko-Pastur law. This removes the higher-cumulant tensor assumption of Fan, Ma, Paquette, and Wang (2026), thereby answering the question raised in their work. The result applies, among other examples, to every centered log-concave column distribution with bounded, nondegenerate covariance, nonlinear tilts of Gaussian vectors, deep random features, and a high-temperature spherical 4-spin model for which the cumulant assumption is known to fail.
Related
- Tracy Widom law for sample covariance matrices under quadratic form concentration
- Quantitative Resolvent Calculus for Sample Covariance Matrices
- A Law of Fractional Logarithm for Nested Complex Sample Covariance Matrices
- Limiting moments of sample covariance matrices from large fold tensor vectors
- On Eigenvector Statistics of Large-Dimensional Sample Covariance Matrices with Applications
BibTeX
@misc{ma2026anisotropic,
title = {{The anisotropic local law for sample covariance matrices under quadratic-form concentration}},
author = {Renyuan Ma and Theodor Misiakiewicz},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09440},
eprint = {2609.09440},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09440},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The anisotropic local law for sample covariance matrices under quadratic-form concentration
AU - Renyuan Ma
AU - Theodor Misiakiewicz
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.09440
UR - https://arxiv.org/abs/2609.09440
AB - We study sample covariance matrices K = 1/N ∑_i=1^N x_i x_i^* ∈ ℝ^n × n in the proportional regime n asymp N. The columns $ \mathbf{x}_1, \ldots, \mathbf{x}_N \in \mathbb{R}^n$ are independent and centered, with common covariance 𝔼 x_i x_i^* = Σ, but may otherwise have strongly and nonlinearly dependent coordinates. Assuming only that quadratic forms of the columns concentrate uniformly at the optimal rate | x_i^* A x_i − Tr ΣA | ≺ | A |_F, together with polynomial norm moments and a standard nondegeneracy condition on Σ, we prove the optimal anisotropic local law: on regular spectral domains, uniformly down to spectral scales η:= Im z ≥ N^−1 + τ, | ⟨ u , ( (K−z)^−1 − (−zI_n−zm̃_0(z)Σ)^−1 ) v > | ≺ √((Im m̃_0 (z))/Nη) + 1/Nη for all deterministic unit vectors $ \mathbf{u}, \mathbf{v} \in \mathbb{C}^n$, where m̃_0(z) is the Stieltjes transform of the deformed Marchenko-Pastur law. This removes the higher-cumulant tensor assumption of Fan, Ma, Paquette, and Wang (2026), thereby answering the question raised in their work. The result applies, among other examples, to every centered log-concave column distribution with bounded, nondegenerate covariance, nonlinear tilts of Gaussian vectors, deep random features, and a high-temperature spherical 4-spin model for which the cumulant assumption is known to fail.
ER -