Tracy Widom law for sample covariance matrices under quadratic form concentration

Guangchen Chen; Xing Lyu

AI summary · arXiv: 2610.41416v1 · math.PR

Tracy Widom law for sample covariance matrices under quadratic form concentration

Guangchen Chen, Xing Lyu

Original paper Submitted to arXiv on 30 Oct. 2026 License: CC0 1.0 Also in: fr

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The essentials

Under quadratic-form concentration, sample covariance matrices with dependent coordinates have a universal largest-eigenvalue fluctuation matching the Gaussian case and obey the Tracy–Widom law.

Question

Does edge universality for sample covariance matrices hold when coordinates within each observation may be dependent, assuming only concentration of quadratic forms? The work also addresses whether the unconditionality restriction in earlier results for log-concave ensembles can be removed.

Method

The authors study H = N⁻¹∑ᵢ qᵢqᵢᵀ for independent, centered, isotropic vectors in the proportional regime M/N → y∞ ∈ (0,∞) \ {1}, assuming uniform Lᵖ concentration of quadratic forms for every fixed p ≥ 2. They use a local Marchenko–Pastur law and spectral rigidity, then compare the original matrix with a Gaussian-divisible ensemble using Dyson Brownian motion and Lindeberg replacement; the interpolation time is t₀ = N⁻¹ᐟ³⁺ᵋ. A self-consistent equation controls the fourth-cumulant terms in the Green-function comparison.

Findings

Result The largest-eigenvalue fluctuations of H and the Gaussian Wishart matrix W agree after edge centering and N²ᐟ³ scaling.
See the passage in the paper
Suppose that Assumption 1.1 holds, then $\displaystyle\lim_{N\to\infty}\mathbb{P}^{H}\left(N^{2/3}(\lambda_{1}(H)-\lambda_{+})\leq x\right)=\lim_{N\to\infty}\mathbb{P}^{W}\left(N^{2/3}(\lambda_{1}(W)-\lambda_{+})\leq x\right),$ where $\lambda_{+}=(1+\sqrt{y})^{2}$.
Result With the stated centering and scaling, the largest eigenvalue converges to the Tracy–Widom distribution.
See the passage in the paper
Under the same condition as Theorem 1.2, we have $\displaystyle\frac{N\lambda_{1}(H)-(\sqrt{M}+\sqrt{N})^{2}}{(\sqrt{M}+\sqrt{N})(1/\sqrt{M}+1/\sqrt{N})^{1/3}}\to\mathrm{TW}_{1},$ where $\mathrm{TW}_{1}$ denotes the Tracy-Widom distribution.
Authors’ interpretation The result allows dependent coordinates within each vector and does not require the unconditionality assumption used in the cited log-concave edge-universality result.
See the passage in the paper
In particular, our result removes the unconditionality assumption in existing edge universality results for log-concave ensembles [6], resolving the question that originally motivated our work.
Authors’ interpretation The authors state that the quadratic-form concentration assumption extends the result beyond the log-concave class.
See the passage in the paper
The concentration assumption also allows us to extend the result beyond the log-concave class.

Limitations

The theorem is restricted to identity population covariance; extending it to general population covariance is left for future work.
See the passage in the paper
We expect our approach can be extended to the general population case, i.e., $\Sigma^{\frac{1}{2}}H\Sigma^{\frac{1}{2}}$. Such an extension will be considered in a future update.
The proportional-aspect-ratio regime excludes the limiting case y∞ = 1.
See the passage in the paper
We assume that $M/N\equiv y\longrightarrow y^{\infty}\in(0,\infty)\setminus\{1\}$, as $N\to\infty$.

Why it matters

The authors argue that quadratic-form concentration suffices for largest-eigenvalue edge universality in this setting, removing the unconditionality restriction for the cited log-concave result and covering some non-log-concave ensembles. This broadens the class of dependent-coordinate data models for which Tracy–Widom fluctuations can be established.

Written by AI from the paper. Passages are checked word for word against the text we analysed; the rest may contain errors.

AI-generated keywords: Sample covariance matrices Tracy–Widom fluctuations quadratic-form concentration Green function comparison Dyson Brownian motion

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Comprehensive Summary
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Original abstract

The paper studies fluctuations of the largest eigenvalue of a high-dimensional sample covariance matrix when coordinates within each observation may be dependent. The data consist of independent, identically distributed random vectors that are centered and isotropic, meaning they have mean zero and identity covariance. The number of coordinates and observations grow proportionally, with their ratio tending to a positive constant different from one. The central question is whether the largest eigenvalue exhibits the universal Tracy–Widom fluctuations familiar from Gaussian data, without requiring independent coordinates or an unconditional distribution. The main assumption is concentration of quadratic forms: for every fixed moment order, the $L^p$ norm of the deviation of $q^\top A q$ from $\operatorname{Tr} A$ is bounded by a constant times the Hilbert–Schmidt norm of $A$, uniformly over deterministic symmetric matrices $A$. This condition holds for Gaussian and log-concave vectors and, as the authors note, also covers some distributions outside the log-concave class. Under this assumption, the main theorem shows that the largest eigenvalue has the same limiting edge fluctuations as in the corresponding Gaussian Wishart model. The edge is located at $(1+\sqrt{d/N})^2$, where $d$ is the dimension and $N$ is the sample size, and the fluctuation scale is $N^{-2/3}$. With the paper’s finite-size centering and scaling, the largest eigenvalue converges in distribution to the real Tracy–Widom law, denoted $\mathrm{TW}_1$. The proof has two main stages. First, the authors add a small Gaussian component to the data matrix through a matrix-valued interpolation, run for time $N^{-1/3+\varepsilon}$, where $\varepsilon>0$ is small. They use local Marchenko–Pastur laws and eigenvalue rigidity to establish sufficient regularity near the spectral edge. A Dyson Brownian motion result then yields the Tracy–Widom limit for the Gaussian-smoothed matrix. Second, they compare the smoothed matrix with the original matrix by replacing columns one at a time and estimating changes in suitable resolvent, or Green function, statistics. Taylor expansion reduces the comparison to a small number of terms involving quadratic forms. Isotropy cancels the leading contribution, while the remaining terms involve fourth cumulants and higher-order errors. The main technical challenge is to control the cumulant terms more precisely than basic resolvent bounds permit. To do so, the authors remove an additional column, expand the resolvents using the Sherman–Morrison formula, and derive a self-consistent equation for weighted fourth-cumulant contractions. Solving this equation gives improved estimates, making the cumulative column-by-column comparison small enough to transfer the edge law. In particular, the Green function comparison bounds the difference between the relevant smoothed expectations by order $N^{-1/6}$, up to small losses in the exponent. The result removes an unconditionality assumption used in earlier edge-universality work for log-concave ensembles, addressing the motivation stated by the authors. It also extends the conclusion to vectors satisfying their quadratic-form concentration condition. The scope is specific: the columns must be independent and isotropic, the dimension-to-sample-size ratio must stay away from one, and the result concerns the largest eigenvalue at the right edge. The paper does not establish the corresponding result for a general population covariance matrix, identifying that extension as future work.

When dependent coordinates still produce universal eigenvalue fluctuations

Large datasets often contain many measurements collected from each observation. The coordinates within an observation may be related, yet researchers still want to understand the overall patterns in the data. One way to study those patterns is through the eigenvalues of a sample covariance matrix. In particular, the largest eigenvalue describes the strongest direction of variation—and its fluctuations can reveal whether a dataset behaves like a familiar random model.

In “Tracy Widom law for sample covariance matrices under quadratic form concentration,” Guangchen Chen and Xing Lyu ask whether the largest eigenvalue follows the same universal fluctuation law as it does for Gaussian data, even when the coordinates of each observation are dependent. Their answer is yes under a broad concentration assumption, without requiring the vectors to have an unconditional distribution.

The setting: many observations, many coordinates

The paper considers independent, identically distributed random vectors. Each vector is centered, so its mean is zero, and isotropic, meaning its covariance matrix is the identity. The dimension and the number of observations both grow, with their ratio approaching a positive constant different from one.

The coordinates within a single vector need not be independent. Instead, the key requirement controls how much a quadratic form can fluctuate. For any fixed moment order and any deterministic symmetric matrix, the deviation of the quadratic form from the matrix’s trace must have an Lp size bounded by a constant times the matrix’s Hilbert–Schmidt norm. In everyday terms, this condition says that the vectors do not produce unusually large fluctuations when viewed through a broad class of weighted sums.

Gaussian vectors satisfy this condition, as do log-concave vectors. The authors also note that it reaches some distributions outside the log-concave class, making the assumption useful beyond those two familiar settings.

The main result: the Gaussian edge law survives

Under quadratic-form concentration, the largest eigenvalue has the same limiting edge fluctuations as in the corresponding Gaussian Wishart model. The edge lies near (1 + √(d/N))², where d is the dimension and N is the number of observations. The fluctuations around that edge occur on the scale N−2/3.

Using the paper’s finite-size centering and scaling, the largest eigenvalue converges in distribution to the real Tracy–Widom law, denoted TW1. This is a striking form of universality: the precise distribution of each observation can differ from the Gaussian case, and its coordinates can be dependent, yet the largest eigenvalue retains the same characteristic edge behavior.

How the proof works

The argument has two stages. First, the authors add a small Gaussian component to the data matrix through an interpolation. They run this smoothing procedure for a time of N−1/3+ε, with ε positive and small. Local Marchenko–Pastur laws and eigenvalue rigidity show that the smoothed matrix has enough regularity near the spectral edge. A result for Dyson Brownian motion then gives the Tracy–Widom limit for this Gaussian-smoothed matrix.

The second stage connects that result back to the original, unsmoothed data. The authors replace columns one at a time and compare suitable resolvent statistics, also called Green functions. Taylor expansion reduces the comparison to a small number of terms involving quadratic forms. Isotropy cancels the leading contribution, but fourth-cumulant terms and higher-order errors remain.

Controlling the difficult terms

Basic resolvent estimates are not strong enough to handle the cumulant terms. The authors address this by removing an additional column and expanding the resolvent with the Sherman–Morrison formula. This leads to a self-consistent equation for weighted fourth-cumulant contractions. Solving that equation gives sharper estimates and keeps the accumulated error from replacing all the columns small.

In particular, the Green function comparison bounds the difference between the relevant smoothed expectations by order N−1/6, up to small losses in the exponent. That comparison is the bridge that transfers the edge law from the smoothed matrix to the original one.

Why the result matters—and where it stops

The work removes an unconditionality assumption used in earlier edge-universality results for log-concave ensembles. Its conclusion also applies to vectors beyond the log-concave class whenever they satisfy the stated quadratic-form concentration condition. This gives researchers a way to establish Gaussian-style largest-eigenvalue behavior without assuming coordinate independence.

The result has a defined scope. The observations must be independent and isotropic, the dimension-to-sample-size ratio must stay away from one, and the theorem concerns the largest eigenvalue at the right edge. It does not establish the same conclusion for a general population covariance matrix; the authors identify that extension as future work.

Within these boundaries, the message is clear: independence among coordinates is not always necessary for universal edge fluctuations. Strong enough control of quadratic forms can preserve the familiar Tracy–Widom law even when observations have a more complicated internal structure.

Created on 02 Nov. 2026

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Weekly letter Presented in "Mathematics and statistics this week" of 2 November 2026. Mathematics and statistics this week: Testing the limits of familiar rules

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Tracy Widom law for sample covariance matrices under quadratic form concentration

The essentials

Question

Method

Findings

Limitations

Why it matters

Read further

Definitions

When dependent coordinates still produce universal eigenvalue fluctuations

The setting: many observations, many coordinates

The main result: the Gaussian edge law survives

How the proof works

Controlling the difficult terms

Why the result matters—and where it stops

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