Lucas Benigni

This work establishes a weak mesoscopic local law and quantitative eigenvector delocalization for the Sachdev–Ye–Kitaev Hamiltonian. For even \(q\ll N^{1/2}\), the normalized Stieltjes transform approaches the standard Gaussian law down to scales of order \(qN^{-1/2}\), up to logarithmic factors, both on the full Hilbert space and within each fermion-parity sector. Consequences include eigenvalue-counting and spectral-form-factor estimates, an averaged inverse-participation-ratio bound, and—for fixed \(q\)—high-probability \(\ell^\infty\) delocalization of individual bulk eigenvectors in any deterministic basis.

A new analysis of the Dyson vector flow yields quantitative eigenvector universality for generalized Wigner matrices without relying on the eigenvector moment flow. The results give joint asymptotic normality for eigenvector projections throughout the spectrum and a quantitative lower bound for an eigenvector’s largest coordinate. When the matrix entries are smooth, the argument handles an explicitly growing collection of projections and provides a convergence rate in Kolmogorov distance.

This paper studies stochastic spectral optimizers such as Muon through a high-dimensional matrix-valued least-squares model. Explicit deterministic dynamics reveal that SignSVD, which Muon approximates, produces square-root preconditioning at large batch size, while small spectral modes revert toward slower SGD-like behavior at small batch size. SignSGD, used as a proxy for Adam, generally provides no comparable covariance preconditioning. For power-law data and target spectra, the comparison produces three regimes: one favoring SignSGD, one favoring SignSVD, and one exhibiting a performance trade-off.

For orthogonally invariant real random matrices, eigenvectors become more localized as their eigenvalues approach the real axis. Localization is measured through the inverse participation ratio. Conditioning on an eigenvalue whose imaginary part is \(y/\sqrt N\), the associated eigenvector’s IPR converges to an explicit random variable \(\ell_y\). Its limiting behavior interpolates between \(3\) near the real axis and \(2\) far from it. The result extends to higher-order IPRs and to the real elliptic Ginibre ensemble across all nonsymmetric parameters.

The paper determines the limiting empirical eigenvalue distribution of a Hadamard product of \(k\) independent sample covariance matrices. Each data matrix has independent rows while allowing a general correlation structure within every row. The asymptotic law is computed in the high-dimensional regime where the sample size \(n\) and feature dimensions \(d_1,\ldots,d_k\) satisfy \(n/(d_1\cdots d_k)\to\gamma\).

This work computes the asymptotic eigenvalue distribution of the neural tangent kernel for a two-layer network in a quadratic high-dimensional scaling. With \(n/(dp)\to\gamma_1\) and \(p/d\to\gamma_2\), the limiting distribution is described as the free multiplicative convolution of a Marchenko–Pastur law with a deterministic measure determined by the activation derivative and the diagonal weight matrix. The result covers pseudo-Lipschitz activation derivatives and bounded random diagonal weights.

For two independent data matrices with i.i.d. entries, this note analyzes the entrywise product of their sample covariance matrices. In the quadratic regime \(n/(dp)\to\gamma\), with \(p/d\to a\), the empirical spectral distribution of \((XX^\top/d)\odot(YY^\top/p)\) converges to the Marchenko–Pastur distribution with shape parameter \(\gamma\).

The paper proves convergence of the processes \((\sqrt N\langle u_k,A_tu_k\rangle)_{t\in[0,1]}\), where \(u_k\) is a bulk eigenvector of a generalized Wigner matrix and \((A_t)\) is a Hölder-regular family of bounded symmetric observables. It identifies explicit limiting processes and shows that a broad class of Gaussian processes with Hölder-continuous covariance can arise through a Karhunen–Loève construction. The proof combines multidimensional eigenvector convergence with a tightness argument derived from the observables’ regularity.

Bessel fields arise as hard-edge scaling limits of the Laguerre field and have Bessel point-process marginals. This paper uncovers additional integrable structure: restricting the field to a time-like or space-like path produces a determinantal point process with an explicit correlation kernel. At any fixed time, varying the Bessel index instead yields an exponential Gibbsian line ensemble.

This work studies how eigenvectors associated with spectral-edge eigenvalues distribute their mass over a macroscopic set of coordinates. After suitable centering and rescaling, that mass converges to a Gaussian as the matrix size grows. The proof uses two-moment matching to compare edge-eigenvector observables for a general Wigner matrix directly with the corresponding quantities in a Gaussian ensemble, where they can be evaluated explicitly.

Any finite family of quadratic forms built from deterministic matrices and eigenvectors of a Wigner matrix is shown to have joint Gaussian fluctuations. These observables describe eigenvector overlaps and provide a random-matrix counterpart of Berry’s random-wave conjecture, connecting the fine statistics of Wigner eigenvectors with Gaussian-wave behavior.

The largest eigenvalues of nonlinear covariance matrices arising from single-layer random neural networks are analyzed. For \(Y=f(WX)\), the conjugate kernel \(YY^\top/m\) has a top eigenvalue whose probability limit matches that of an associated linear information-plus-noise ensemble. Depending on the activation \(f\) and the distributions of \(W\) and \(X\), the model can exhibit a phase transition, linking nonlinear random-matrix behavior to questions in machine learning.

For generalized Wigner matrices, the eigenvector mass carried by mesoscopic coordinate sets of size between \(N^\varepsilon\) and \(N^{1-\varepsilon}\) is proved to converge to a Gaussian at every spectral energy, including the edge. A four-point eigenvector decorrelation estimate is obtained through new moment observables governed by parabolic equations and a maximum principle. A bootstrap argument also gives high-probability quantum unique ergodicity and quantum weak-mixing bounds simultaneously for all eigenvectors and deterministic coordinate sets.

Generalized Wigner eigenvectors with subexponential entries are shown to delocalize at the optimal rate with overwhelming probability, together with sharp-constant high-probability bounds. The proof first controls logarithmic moments through the eigenvector moment flow for matrices containing a small Gaussian component, then removes that component using comparison arguments based on regularized eigenvectors and level repulsion. The analysis also establishes spectrum-wide level-repulsion and eigenvalue-overcrowding estimates.

This paper determines the limiting empirical spectrum of the nonlinear random matrix \(YY^*/m\) with \(Y=f(WX)\), a model for random neural-network features. The matrices \(W\) and \(X\) have centered i.i.d. entries with sub-Gaussian tails, while the activation \(f\) is real analytic and applied entrywise. The result extends earlier Gaussian-input calculations and also treats analogous spectral questions for multilayer networks.

New fermionic observables of Dyson Brownian-motion eigenvectors are introduced and shown to satisfy an evolution analogous to the Bourgade–Yau eigenvector moment flow. Combining these fermionic quantities with the original bosonic observables reveals new eigenvector correlations. For generalized Wigner matrices, mass fluctuations associated with distinct eigenvectors decorrelate as \(N\) grows, and the method gives an optimal estimate for partial inner products between different eigenvectors.

The eigenvectors of mesoscopic mean-field perturbations of diagonal matrices are analyzed through a generalized Rosenzweig–Porter model. Their entries become asymptotically Gaussian with an explicit variance profile that confines mass to a small spectral region; for well-spread initial spectra, this profile has a universal heavy-tailed Cauchy form. With smooth entries, the paper also proves a strong, overwhelming-probability version of quantum unique ergodicity using local laws and the eigenvector moment flow.

For fractional Brownian motion with Hurst parameter \(H\), the record set consists of times at which the process reaches a new running maximum. This paper proves that the Hausdorff dimension of that random set is almost surely equal to \(H\).