Related papers: Sharp estimates for large N Weingarten functions
We prove a functional limit theorem for vector-valued functionals of the fractional Ornstein-Uhlenbeck process, providing the foundation for the fluctuation theory of slow/fast systems driven by such a noise. Our main contribution is on the…
A method for computing integrals of polynomial functions on compact symmetric spaces is given. Those integrals are expressed as sums of functions on symmetric groups.
Some eigenvalue matrix models possess an interesting property: one can manifestly define the basis where all averages can be explicitly calculated. For example, in the Gaussian Hermitian and rectangular complex models, averages of the Schur…
Wigner functions generically attain negative values and hence are not probability densities. We prove an asymptotic expansion of Wigner functions in terms of Hermite spectrograms, which are probability densities. The expansion provides…
We prove a Chernoff-type bound for sums of matrix-valued random variables sampled via a regular (aperiodic and irreducible) finite Markov chain. Specially, consider a random walk on a regular Markov chain and a Hermitian matrix-valued…
Let $U^N$ be a family of $N\times N$ independent Haar unitary random matrices and their adjoints, $Z^N$ a family of deterministic matrices, and $P$ a self-adjoint noncommutative polynomial, i.e. for any $N$, $P(U^N,Z^N)$ is self-adjoint,…
The Wasserstein metric is broadly used in optimal transport for comparing two probabilistic distributions, with successful applications in various fields such as machine learning, signal processing, seismic inversion, etc. Nevertheless, the…
The current work applies some recent combinatorial tools due to Jain to control the eigenvalue gaps of a matrix $M_n = M + N_n$ where $M$ is deterministic, symmetric with large operator norm and $N_n$ is a random symmetric matrix with…
We prove convergence of eigenvector processes of the form $(\sqrt{N}\langle \mathbf{u}_k,A_t\mathbf{u}_k\rangle)_{t\in[0,1]}$ where $\mathbf{u}_k$ is a bulk eigenvector of generalized Wigner matrices and $(A_t)$ a family of symmetric…
Unitary ensembles of large N x N random matrices with a non-Gaussian probability distribution P[H] ~ exp{-TrV[H]} are studied using a theory of polynomials orthogonal with respect to exponential weights. Asymptotically exact expressions for…
The past thirteen years have seen the development of many algorithms for approximating matrix functions in O(N) time, where N is the basis size. These O(N) algorithms rely on assumptions about the spatial locality of the matrix function;…
With the ansatz that a data set's correlation matrix has a certain parametrized form (one general enough, however, to allow the arbitrary specification of a slowly-varying decorrelation distance and population variance) the general…
Understanding the limiting behavior of eigenvalues of random matrices is the central problem of random matrix theory. Classical limit results are known for many models, and there has been significant recent progress in obtaining more…
We study the problem of estimating functions of a large symmetric matrix $A_n$ when we only have access to a noisy estimate $\hat{A}_n=A_n+\sigma Z_n/\sqrt{n}.$ We are interested in the case that $Z_n$ is a Wigner ensemble and suggest an…
Due to instanton effects, gauge-theoretic large N expansions yield asymptotic series, in powers of 1/N^2. The present work shows how to generically make such expansions meaningful via their completion into resurgent transseries, encoding…
A set of functions is defined which is indexed by a positive integer $n$ and partitions of integers. The case $n=1$ reproduces the standard Schur polynomials. These functions are seen to arise naturally as a determinant of an action on the…
There have been several algorithms designed to optimise matrix multiplication. From schoolbook method with complexity $O(n^3)$ to advanced tensor-based tools with time complexity $O(n^{2.3728639})$ (lowest possible bound achieved), a lot of…
We prove various theorems on approximation using polynomials with integer coefficients in the Bernstein basis of any given order. In the extreme, we draw the coefficients from $\{ \pm 1\}$ only. A basic case of our results states that for…
Matrix multiplication is a fundamental task in almost all computational fields, including machine learning and optimization, computer graphics, signal processing, and graph algorithms (static and dynamic). Twin-width is a natural complexity…
We compute the deterministic approximation of products of Sobolev functions of large Wigner matrices $W$ and provide an optimal error bound on their fluctuation with very high probability. This generalizes Voiculescu's seminal theorem…