arXiv subjects

Kevin Schnelli

Publications and source records attributed to Kevin Schnelli.

At least 19 recordsLinked to original sources

arXiv2026

Order statistics for edge eigenvectors of Wigner matrices

In this paper, we establish a general comparison theorem for the order statistics of the edge eigenvectors for generalized Wigner matrices. Consequently, we derive the Gumbel law for the maximal edge eigenvector component and prove the universality of the Gaussian fluctuations of the order statistics in an intermediate regime close to the maximum. In addition, our comparison result also implies a quantitative first order estimate for moderately small order statistics.

math.PR↗

arXiv2025

Quantitative Tracy-Widom laws for sparse random matrices

We consider the fluctuations of the largest eigenvalue of sparse random matrices, the class of random matrices that includes the normalized adjacency matrices of the Erd\H{o}s-R\'enyi graph $G(N, p)$. We show that the fluctuations of the largest eigenvalue converge to the Tracy-Widom law at a rate almost $O(N^{-1/3 } + p^{-2} N^{-4/3})$ in the regime $p \gg N^{-2/3 }$. Our proof builds upon the Green function comparison method initiated by Erd\H{o}s, Yau, and Yin [22]. To show a Green function comparison theorem for fine spectral scales, we implement algorithms for symbolic computations involving averaged products of Green function entries.

math.PR↗

arXiv2024

Asymptotic freeness through unitaries generated by polynomials of Wigner matrices

We study products of functions evaluated at self-adjoint polynomials in deterministic matrices and independent Wigner matrices; we compute the deterministic approximations of such products and control the fluctuations. We focus on minimizing the assumption of smoothness on those functions while optimizing the error term with respect to $N$, the size of the matrices. As an application, we build on the idea that the long-time Heisenberg evolution associated to Wigner matrices generates asymptotic freeness as first shown in $[9]$. More precisely given $P$ a self-adjoint non-commutative polynomial and $Y^N$ a $d$-tuple of independent Wigner matrices, we prove that the quantum evolution associated to the operator $P(Y^N)$ yields asymptotic freeness for large times.

math.PR↗

arXiv2023

The free energy of matrix models

In this paper we study multi-matrix models whose potentials are perturbations of the quadratic potential associated with independent GUE random matrices. More precisely, we compute the free energy and the expectation of the trace of polynomials evaluated in those matrices. We prove an asymptotic expansion in the inverse of the matrix dimension to any order. Out of this result we deduce new formulas for map enumerations and the microstates free entropy. Our approach is based on the interpolation method between random matrices and free operators developed in [9, 31].

math.PR↗

arXiv2022

Quantitative Tracy-Widom laws for the largest eigenvalue of generalized Wigner matrices

We show that the fluctuations of the largest eigenvalue of any generalized Wigner matrix $H$ converge to the Tracy-Widom laws at a rate nearly $O(N^{-1/3})$, as the matrix dimension $N$ tends to infinity. We allow the variances of the entries of $H$ to have distinct values but of comparable sizes such that $\sum_{i} \mathbb{E}|h_{ij}|^2=1$. Our result improves the previous rate $O(N^{-2/9})$ by Bourgade [8] and the proof relies on the first long-time Green function comparison theorem near the edges without the second moment matching restriction.

math.PR↗

arXiv2022

Convergence rate to the Tracy-Widom laws for the largest eigenvalue of Wigner matrices

We show that the fluctuations of the largest eigenvalue of a real symmetric or complex Hermitian Wigner matrix of size $N$ converge to the Tracy--Widom laws at a rate $O(N^{-1/3+ω})$, as $N$ tends to infinity. For Wigner matrices this improves the previous rate $O(N^{-2/9+ω})$ obtained by Bourgade [5] for generalized Wigner matrices. Our result follows from a Green function comparison theorem, originally introduced by Erdős, Yau and Yin [19] to prove edge universality, on a finer spectral parameter scale with improved error estimates. The proof relies on the continuous Green function flow induced by a matrix-valued Ornstein--Uhlenbeck process. Precise estimates on leading contributions from the third and fourth order moments of the matrix entries are obtained using iterative cumulant expansions and recursive comparisons for correlation functions, along with uniform convergence estimates for correlation kernels of the Gaussian invariant ensembles.

math.PR↗

arXiv2022

The support of the free additive convolution of multi-cut measures

We consider the free additive convolution $μ_α\boxplusμ_β$ of two probability measures $μ_α$ and $μ_β$, supported on respectively $n_α$ and $n_β$ disjoint bounded intervals on the real line, and derive a lower bound and an upper bound that is strictly smaller than $2n_αn_β$, on the number of connected components in its support. We also obtain the corresponding results for the free additive convolution semi-group $\{μ^{\boxplus t}\,:\, t\ge 1\}$. Throughout the paper, we consider classes of probability measures with power law behaviors at the endpoints of their supports with exponents ranging from $-1$ to $1$. Our main theorem generalizes a result of Bao, Erdős and Schnelli~[4] to the multi-cut setup.

math.PR↗

arXiv2021

Convergence rate to the Tracy--Widom laws for the largest eigenvalue of sample covariance matrices

We establish a quantitative version of the Tracy--Widom law for the largest eigenvalue of high dimensional sample covariance matrices. To be precise, we show that the fluctuations of the largest eigenvalue of a sample covariance matrix $X^*X$ converge to its Tracy--Widom limit at a rate nearly $N^{-1/3}$, where $X$ is an $M \times N$ random matrix whose entries are independent real or complex random variables, assuming that both $M$ and $N$ tend to infinity at a constant rate. This result improves the previous estimate $N^{-2/9}$ obtained by Wang [73]. Our proof relies on a Green function comparison method [27] using iterative cumulant expansions, the local laws for the Green function and asymptotic properties of the correlation kernel of the white Wishart ensemble.

math.PR↗

arXiv2020

Central limit theorem for mesoscopic eigenvalue statistics of deformed Wigner matrices and sample covariance matrices

We consider $N$ by $N$ deformed Wigner random matrices of the form $X_N=H_N+A_N$, where $H_N$ is a real symmetric or complex Hermitian Wigner matrix and $A_N$ is a deterministic real bounded diagonal matrix. We prove a universal Central Limit Theorem for the linear eigenvalue statistics of $X_N$ for all mesoscopic scales both in the spectral bulk and at regular edges where the global eigenvalue density vanishes as a square root. The method relies on the characteristic function method in [47], local laws for the Green function of $X_N$ in [3, 46, 51] and analytic subordination properties of the free additive convolution [24, 41]. We also prove the analogous results for high-dimensional sample covariance matrices.

math.PR↗

arXiv2020

Central limit theorem for mesoscopic eigenvalue statistics of the free sum of matrices

We consider random matrices of the form $H_N=A_N+U_N B_N U^*_N$, where $A_N$, $B_N$ are two $N$ by $N$ deterministic Hermitian matrices and $U_N$ is a Haar distributed random unitary matrix. We establish a universal Central Limit Theorem for the linear eigenvalue statistics of $H_N$ on all mesoscopic scales inside the regular bulk of the spectrum. The proof is based on studying the characteristic function of the linear eigenvalue statistics, and consists of two main steps: (1) generating Ward identities using the left-translation-invariance of the Haar measure, along with a local law for the resolvent of $H_N$ and analytic subordination properties of the free additive convolution, allow us to derive an explicit formula for the derivative of the characteristic function; (2) a local law for two-point product functions of resolvents is derived using a partial randomness decomposition of the Haar measure. We also prove the corresponding results for orthogonal conjugations.

math.PR↗

arXiv2020

Equipartition principle for Wigner matrices

We prove that the energy of any eigenvector of a sum of several independent large Wigner matrices is equally distributed among these matrices with very high precision. This shows a particularly strong microcanonical form of the equipartition principle for quantum systems whose components are modelled by Wigner matrices.

math-ph↗

arXiv2020

Spectral rigidity for addition of random matrices at the regular edge

We consider the sum of two large Hermitian matrices $A$ and $B$ with a Haar unitary conjugation bringing them into a general relative position. We prove that the eigenvalue density on the scale slightly above the local eigenvalue spacing is asymptotically given by the free convolution of the laws of $A$ and $B$ as the dimension of the matrix increases. This implies optimal rigidity of the eigenvalues and optimal rate of convergence in Voiculescu's theorem. Our previous works [3,4] established these results in the bulk spectrum, the current paper completely settles the problem at the spectral edges provided they have the typical square-root behavior. The key element of our proof is to compensate the deterioration of the stability of the subordination equations by sharp error estimates that properly account for the local density near the edge. Our results also hold if the Haar unitary matrix is replaced by the Haar orthogonal matrix.

math.PR↗

arXiv2019

Local single ring theorem on optimal scale

Let $U$ and $V$ be two independent $N$ by $N$ random matrices that are distributed according to Haar measure on $U(N)$. Let $Σ$ be a non-negative deterministic $N$ by $N$ matrix. The single ring theorem [26] asserts that the empirical eigenvalue distribution of the matrix $X:= UΣV^*$ converges weakly, in the limit of large $N$, to a deterministic measure which is supported on a single ring centered at the origin in $\mathbb{C}$. Within the bulk regime, i.e. in the interior of the single ring, we establish the convergence of the empirical eigenvalue distribution on the optimal local scale of order $N^{-1/2+\varepsilon}$ and establish the optimal convergence rate. The same results hold true when~$U$ and~$V$ are Haar distributed on $O(N)$.

math.PR↗

arXiv2018

On the support of the free additive convolution

We consider the free additive convolution of two probability measures $μ$ and $ν$ on the real line and show that $μ\boxplusν$ is supported on a single interval if $μ$ and $ν$ each has single interval support. Moreover, the density of $μ\boxplusν$ is proven to vanish as a square root near the edges of its support if both $μ$ and $ν$ have power law behavior with exponents between $-1$ and $1$ near their edges. In particular, these results show the ubiquity of the conditions in our recent work on optimal local law at the spectral edges for addition of random matrices [4].

math-ph↗

arXiv2018

Local law and Tracy-Widom limit for sparse sample covariance matrices

We consider spectral properties of sparse sample covariance matrices, which includes biadjacency matrices of the bipartite Erdős-Rényi graph model. We prove a local law for the eigenvalue density up to the upper spectral edge. Under a suitable condition on the sparsity, we also prove that the limiting distribution of the rescaled, shifted extremal eigenvalues is given by the GOE Tracy-Widom law with an explicit formula on the deterministic shift of the spectral edge. For the biadjacency matrix of an Erdős-Rényi graph with two vertex sets of comparable sizes $M$ and $N$, this establishes Tracy-Widom fluctuations of the second largest eigenvalue when the connection probability $p$ is much larger than $N^{-2/3}$ with a deterministic shift of order $(Np)^{-1}$.

math.PR↗

arXiv2016

Convergence Rate for Spectral Distribution of Addition of Random Matrices

Let $A$ and $B$ be two $N$ by $N$ deterministic Hermitian matrices and let $U$ be an $N$ by $N$ Haar distributed unitary matrix. It is well known that the spectral distribution of the sum $H=A+UBU^*$ converges weakly to the free additive convolution of the spectral distributions of $A$ and $B$, as $N$ tends to infinity. We establish the optimal convergence rate ${\frac{1}{N}}$ in the bulk of the spectrum.

math.PR↗

arXiv2016

Bulk universality for deformed Wigner matrices

We consider $N\times N$ random matrices of the form $H=W+V$ where $W$ is a real symmetric or complex Hermitian Wigner matrix and $V$ is a random or deterministic, real, diagonal matrix whose entries are independent of $W$. We assume subexponential decay for the matrix entries of $W$, and we choose $V$ so that the eigenvalues of $W$ and $V$ are typically of the same order. For a large class of diagonal matrices $V$, we show that the local statistics in the bulk of the spectrum are universal in the limit of large $N$.

math.PR↗

arXiv2016

Local law and Tracy-Widom limit for sparse random matrices

We consider spectral properties and the edge universality of sparse random matrices, the class of random matrices that includes the adjacency matrices of the Erdos-Renyi graph model $G(N,p)$. We prove a local law for the eigenvalue density up to the spectral edges. Under a suitable condition on the sparsity, we also prove that the rescaled extremal eigenvalues exhibit GOE Tracy-Widom fluctuations if a deterministic shift of the spectral edge due to the sparsity is included. For the adjacency matrix of the Erdos-Renyi graph this establishes the Tracy-Widom fluctuations of the second largest eigenvalue for $p\gg N^{-2/3}$ with a deterministic shift of order $(Np)^{-1}$.

math.PR↗