Related papers: Smallest eigenvalue distribution of the fixed trac…
We studied numerically the distribution of the entanglement Hamiltonian eigenvalues in two one-dimensional free fermion models and the typical three-dimensional Anderson model. We showed numerically that this distribution depends on the…
The six Painlev\'e transcendants which originally appeared in the studies of ordinary differential equations have been found numerous applications in physical problems. The well-known examples among which include symmetry reduction of the…
We investigate the statistics of the largest eigenvalue, $\lambda_{\rm max}$, in an ensemble of $N\times N$ large ($N\gg 1$) sparse adjacency matrices, $A_N$. The most attention is paid to the distribution and typical fluctuations of…
We study the distribution of the minimum free energy (MFE) for the Turner model of pseudoknot free RNA secondary structures over ensembles of random RNA sequences. In particular, we are interested in those rare and intermediate events of…
We consider the adjacency matrices of sparse random graphs from the Chung-Lu model, where edges are added independently between the $N$ vertices with varying probabilities $p_{ij}$. The rank of the matrix $(p_{ij})$ is some fixed positive…
We show that some of the best-known matrix decompositions of some of the best-known random matrix ensembles give us the unique $G$-invariant uniform distributions on some of the best-known manifolds. The eigenvectors distributions of the…
Organising the space of entanglement structures of a multipartite quantum system is a much more challenging task than its bipartite version: while the local unitary (LU) orbit of a bipartite pure state can be conveniently characterized by…
In this text, we consider an N by N random matrix X such that all but o(N) rows of X have W non identically zero entries, the other rows having lass than $W$ entries (such as, for example, standard or cyclic band matrices). We always…
We study the spectral norm of random kernel matrices with polynomial scaling, where the number of samples scales polynomially with the data dimension. In this regime, Lu and Yau (2022) proved that the empirical spectral distribution…
We consider the ensemble of $N\times N$ ($N\gg 1$) symmetric random matrices with the bimodal independent distribution of matrix elements: each element could be either "1" with the probability $p$, or "0" otherwise. We pay attention to the…
We consider the eigenvalues of sample covariance matrices of the form $\mathcal{Q}=(\Sigma^{1/2}X)(\Sigma^{1/2}X)^*$. The sample $X$ is an $M\times N$ rectangular random matrix with real independent entries and the population covariance…
We study the problem of partitioning a small sample of $n$ individuals from a mixture of $k$ product distributions over a Boolean cube $\{0, 1\}^K$ according to their distributions. Each distribution is described by a vector of allele…
Finite sample properties of random covariance-type matrices have been the subject of much research. In this paper we focus on the "lower tail" of such a matrix, and prove that it is subgaussian under a simple fourth moment assumption on the…
We study the singular values of certain triangular random matrices. When their elements are i.i.d. standard complex Gaussian random variables, the squares of the singular values form a biorthogonal ensemble, and with an appropriate change…
We study the angles between the eigenvectors of a random $n\times n$ complex matrix $M$ with density $\propto \mathrm{e}^{-n\operatorname{Tr}V(M^*M)}$ and $x\mapsto V(x^2)$ convex. We prove that for unit eigenvectors…
The distribution function for the first eigenvalue spacing in the Laguerre unitary ensemble of finite size may be expressed in terms of a solution of the fifth Painleve transcendent. The generating function of a certain discontinuous linear…
In this work, we present a method to exponentiate non-sparse indefinite low-rank matrices on a quantum computer. Given an operation for accessing the elements of the matrix, our method allows singular values and associated singular vectors…
The density matrices are positively semi-definite Hermitian matrices of unit trace that describe the state of a quantum system. The goal of the paper is to develop minimax lower bounds on error rates of estimation of low rank density…
Learning generative probabilistic models is a core problem in machine learning, which presents significant challenges due to the curse of dimensionality. This paper proposes a joint dimensionality reduction and non-parametric density…
We consider the beta-Laguerre ensemble, a family of distributions generalizing the joint eigenvalue distribution of the Wishart random matrices. We show that the bulk scaling limit of these ensembles exists for all beta>0 for a general…