The anisotropic local law for sample covariance matrices under quadratic-form concentration

Abstract

We study sample covariance matrices $K=\frac{1}{N}{\sum }_{i=1}^N{\mathbf{x}}_i{\mathbf{x}}_i^{\ast }\in {\mathbb{R}}^{n\times n}$ in the proportional regime $n\asymp N$. The columns ${\mathbf{x}}_1,\dots ,{\mathbf{x}}_N\in {\mathbb{R}}^n$ are independent and centered, with common covariance $\mathbb{E}{\mathbf{x}}_i{\mathbf{x}}_i^{\ast }=\Sigma$, but may otherwise have strongly and nonlinearly dependent coordinates. Assuming only that quadratic forms of the columns concentrate uniformly at the optimal rate $|{\mathbf{x}}_i^{\ast }A{\mathbf{x}}_i-\mathrm{Tr}\Sigma A|\prec \Vert A{\Vert }_F$, together with polynomial norm moments and a standard nondegeneracy condition on $\Sigma$, we prove the optimal anisotropic local law: on regular spectral domains, uniformly down to spectral scales $\eta :=\mathrm{Im}z\ge N^{-1+\tau }$, \[ \big| \langle \mathbf{u} , \big( (K-z)^{-1} - (-zI_n-z\widetilde m_0(z)Σ\big)^{-1} \big) \mathbf{v} \> \big| \prec \sqrt{\frac{\mathrm{Im}\, \widetilde m_0 (z)}{Nη}} + \frac{1}{Nη} \] for all deterministic unit vectors $\mathbf{u},\mathbf{v}\in {\mathbb{C}}^n$, where ${\tilde{m}}_0(z)$ 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.