Extreme Eigenvalues Research Articles

On a new functional for extremal metrics of the conformal Laplacian in high dimensions

In this paper, we introduce a new functional for the conformal spectrum of the conformal Laplacian on a closed manifold [Formula: see text] of dimension at least 3. For this new functional we provide a Korevaar type result. The main body of the paper deals with the case of the sphere but a section is devoted to more general closed manifolds. The functional introduced here has been investigated in the recent work [S. Perez-Ayala, Extremal eigenvalues of the conformal Laplacian under Sire–Xu normalization, preprint (2019); arXiv:2011.06018].

Shape Optimization of a Weighted Two-Phase Dirichlet Eigenvalue

Let $m$ be a bounded function and $\alpha$ a nonnegative parameter. This article is concerned with the first eigenvalue $\lambda\_\alpha(m)$ of the drifted Laplacian type operator $\mathcal L\_m$ given by $\mathcal L\_m(u)= -\operatorname{div} \left((1+\alpha m)\nabla u\right)-mu$ on a smooth bounded domain, with Dirichlet boundary conditions. Assuming uniform pointwise and integral bounds on $m$, we investigate the issue of minimizing $\lambda\_\alpha(m)$ with respect to $m$. Such a problem is related to the so-called "two phase extremal eigenvalue problem" and arises naturally, for instance in population dynamics where it is related to the survival ability of a species in a domain. We prove that unless the domain is a ball, this problem has no "regular" solution. We then provide a careful analysis in the case of a ball by: (1) characterizing the solution among all radially symmetric resources distributions, with the help of a new method involving a homogenized version of the problem; (2) proving in a more general setting, a stability result for the centered distribution of resources with the help of a monotonicity principle for second order shape derivatives which significantly simplifies the analysis.

Extreme eigenvalue statistics of m-dependent heavy-tailed matrices

Nous analysons les plus grandes valeurs propres d’une matrice de Wigner avec entrées m-dépendantes et à queue lourde, de même que pour une matrice de covariance associée avec entrées de variation régulière de paramètre α∈(0,4). Notre analyse étend les résultats existants pour ces matrices aléatoires avec entrées indépendantes à des entrées m-dépendantes. Nous prouvons que le processus ponctuel limite des plus grandes valeurs propres est un processus de Poisson groupé.

Extremal eigenvalues of sample covariance matrices with general population

We consider the eigenvalues of sample covariance matrices of the form Q=(Σ1/2X)(Σ1/2X)∗. The sample X is an M×N rectangular random matrix with real independent entries and the population covariance matrix Σ is a positive definite diagonal matrix independent of X. Assuming that the limiting spectral density of Σ exhibits convex decay at the right edge of the spectrum, in the limit M,N→∞ with N/M→d∈(0,∞), we find a certain threshold d+ such that for d>d+ the limiting spectral distribution of Q also exhibits convex decay at the right edge of the spectrum. In this case, the largest eigenvalues of Q are determined by the order statistics of the eigenvalues of Σ, and in particular, the limiting distribution of the largest eigenvalue of Q is given by a Weibull distribution. In case d<d+, we also prove that the limiting distribution of the largest eigenvalue of Q is Gaussian if the entries of Σ are i.i.d. random variables. While Σ is considered to be random mostly, the results also hold for deterministic Σ with some additional assumptions.

Tracy–Widom method for Jánossy density and joint distribution of extremal eigenvalues of random matrices

Abstract The Jánossy density for a determinantal point process is the probability density that an interval $I$ contains exactly $p$ points except for those at $k$ designated loci. The Jánossy density associated with an integrable kernel $\mathbf{K}\doteq (\varphi(x)\psi(y)-\psi(x)\varphi(y))/(x-y)$ is shown to be expressed as a Fredholm determinant $\mathrm{Det}(\mathbb{I}-\tilde{\mathbf{K}}|_I)$ of a transformed kernel $\tilde{\mathbf{K}}\doteq (\tilde{\varphi}(x)\tilde{\psi}(y)-\tilde{\psi}(x)\tilde{\varphi}(y))/(x-y)$. We observe that $\tilde{\mathbf{K}}$ satisfies Tracy and Widom’s criteria if $\mathbf{K}$ does, because of the structure that the map $(\varphi, \psi)\mapsto (\tilde{\varphi}, \tilde{\psi})$ is a meromorphic $\mathrm{SL}(2,\mathbb{R})$ gauge transformation between covariantly constant sections. This observation enables application of the Tracy–Widom method [7] to Jánossy densities, expressed in terms of a solution to a system of differential equations in the endpoints of the interval. Our approach does not explicitly refer to isomonodromic systems associated with Painlevé equations employed in the preceding works. As illustrative examples we compute Jánossy densities with $k=1, p=0$ for Airy and Bessel kernels, related to the joint distributions of the two largest eigenvalues of random Hermitian matrices and of the two smallest singular values of random complex matrices.

Markov-type inequalities and extreme zeros of orthogonal polynomials

Let {pm}m=0∞ be an orthogonal system in a Hilbert space H, πn=Span{p0,p1,…,pn}, and L a linear operator defined on H. Under certain assumptions on L, we prove that {cn−2(L)}n∈N, the reciprocals of the squared sharp constants {cn(L)}n∈N in the inequality ‖Lf‖≤cn(L)‖f‖,f∈πn, are equal to the smallest eigenvalues of certain positive definite Jacobi matrices or, equivalently, to the smallest zeros of certain polynomials, orthogonal with respect to a measure supported on R+. Hence, the problem of estimating the sharp constants cn(L) is transformed to the problem of estimating the extreme eigenvalues (zeros) of these Jacobi matrices (orthogonal polynomials). By applying Gershgorin’s Circle Theorem we obtain new upper bounds for the sharp constants in the Markov L2 inequalities with the Laguerre and Gegenbauer weights, and examine the asymptotic behavior of these constants as the parameters in the weights tend to infinity.

Identification of the Majorana edge modes in tight-binding systems based on the Krylov method

Low dimensional structures in the non-trivial topological phase can host the in-gap Majorana bound states, identified experimentally as zero-bias peaks in differential conductance. Theoretical methods for studying Majorana modes are mostly based on the bulk-boundary correspondence or exact diagonalization of finite systems via, e.g., Bogoliubov–de Gennes formalism. In this paper, we develop an efficient method for identifying the Majorana in-gap (edge) states via looking for extreme eigenvalues of symmetric matrices. The presented approach is based on the Krylov method and allows for study the spatial profile of the modes as well as the spectrum of the system. The advantage of this method is the calculation cost, which shows linear dependence on the number of lattice sites. The latter problem may be solved for very large clusters of arbitrary shape/geometry. In order to demonstrate the efficiency of our approach, we study two- and three-dimensional clusters described by the Kitaev and Rashba models for which we determine the number of Majorana modes and calculate their spatial structures. Additionally, we discuss the impact of the system size on the physical properties of the topological phase of the magnetic nanoisland deposited on the superconducting surface. In this case, we have shown that the eigenvalues of the in-gap states depend on the length of the system edge.

Random Matrix Theory and Its Applications

This article reviews the important ideas behind random matrix theory (RMT), which has become a major tool in a variety of disciplines, including mathematical physics, number theory, combinatorics and multivariate statistical analysis. Much of the theory involves ensembles of random matrices that are governed by some probability distribution. Examples include Gaussian ensembles and Wishart–Laguerre ensembles. Interest has centered on studying the spectrum of random matrices, especially the extreme eigenvalues, suitably normalized, for a single Wishart matrix and for two Wishart matrices, for finite and infinite sample sizes in the real and complex cases. The Tracy–Widom Laws for the probability distribution of a normalized largest eigenvalue of a random matrix have become very prominent in RMT. Limiting probability distributions of eigenvalues of a certain random matrix lead to Wigner’s Semicircle Law and Marc˘enko–Pastur’s Quarter-Circle Law. Several applications of these results in RMT are described in this article.

Finite size effects and loss of self-averageness in the relaxational dynamics of the spherical Sherrington–Kirkpatrick model

We revisit the gradient descent dynamics of the spherical Sherrington–Kirkpatrick (p = 2) model with finite number of degrees of freedom. For fully random initial conditions we confirm that the relaxation takes place in three time regimes: a first algebraic one controlled by the decay of the eigenvalue distribution of the random exchange interaction matrix at its edge in the infinite size limit; a faster algebraic one determined by the distribution of the gap between the two extreme eigenvalues; and a final exponential one determined by the minimal gap sampled in the disorder average. We also analyse the finite size effects on the relaxation from initial states which are almost projected on the saddles of the potential energy landscape, and we show that for deviations scaling as N −ν from perfect alignment the system escapes the initial configuration in a time-scale scaling as ln N after which the dynamics no longer ‘self-averages’ with respect to the initial conditions. We prove these statements with a combination of analytic and numerical methods.

Bounds for the extremal eigenvalues of gain Laplacian matrices

A complex unit gain graph (T-gain graph), Φ=(G,φ) is a graph where the function φ assigns a unit complex number to each orientation of an edge of G, and its inverse is assigned to the opposite orientation. A T-gain graph Φ is balanced if the product of the edge gains of each cycle (with a fixed orientation) is 1. Signed graphs are special cases of T-gain graphs.The adjacency matrix of Φ, denoted by A(Φ) is defined canonically. The gain Laplacian for Φ is defined as L(Φ)=D(Φ)−A(Φ), where D(Φ) is the diagonal matrix with diagonal entries are the degrees of the vertices of G. The minimum number of vertices (resp., edges) to be deleted from Φ in order to get a balanced gain graph is the frustration number (resp, frustration index). We show that frustration number and frustration index are bounded below by the smallest eigenvalue of L(Φ). We provide several lower and upper bounds for extremal eigenvalues of L(Φ) in terms of different graph parameters such as the number of edges, vertex degrees, and average 2-degrees. The signed graphs are particular cases of the T-gain graphs, all the bounds established in this paper hold for signed graphs. Most of the bounds established here are new for signed graphs. Finally, we perform comparative analysis for all the obtained bounds in the paper with the state-of-the-art bounds available in the literature for randomly generated Erdős-Reýni graphs.Some of the major highlights of our paper are the gain-dependent bounds, limit convergence of the bounds to the extremal eigenvalues, and optimal extremal bounds obtained by posing optimization problems to achieve the best possible bounds.

Extreme eigenvalues of nonlinear correlation matrices with applications to additive models

The maximum correlation of functions of a pair of random variables is an important measure of stochastic dependence. It is known that this maximum nonlinear correlation is identical to the absolute value of the Pearson correlation for a pair of Gaussian random variables or a pair of finite sums of iid random variables. This paper extends these results to pairwise Gaussian vectors and processes, nested sums of iid random variables, and permutation symmetric functions of sub-groups of iid random variables. It also discusses applications to additive regression models.

Doubly stochastic matrices and the quantum channels

Abstract The main object of this paper is to study doubly stochastic matrices with majorization and the Birkhoff theorem. The Perron-Frobenius theorem on eigenvalues is generalized for doubly stochastic matrices. The region of all possible eigenvalues of n-by-n doubly stochastic matrix is the union of regular (n – 1) polygons into the complex plane. This statement is ensured by a famous conjecture known as the Perfect-Mirsky conjecture which is true for n = 1, 2, 3, 4 and untrue for n = 5. We show the extremal eigenvalues of the Perfect-Mirsky regions graphically for n = 1, 2, 3, 4 and identify corresponding doubly stochastic matrices. Bearing in mind the counterexample of Rivard-Mashreghi given in 2007, we introduce a more general counterexample to the conjecture for n = 5. Later, we discuss different types of positive maps relevant to Quantum Channels (QCs) and finally introduce a theorem to determine whether a QCs gives rise to a doubly stochastic matrix or not. This evidence is straightforward and uses the basic tools of matrix theory and functional analysis.

Extremal eigenvalues of critical Erdős–Rényi graphs

We complete the analysis of the extremal eigenvalues of the adjacency matrix A of the Erdős–Rényi graph G(N,d/N) in the critical regime d≍logN of the transition uncovered in (Ann. Inst. Henri Poincaré Probab. Stat. 56 (2020) 2141–2161; Ann. Probab. 47 (2019) 1653–1676), where the regimes d≫logN and d≪logN were studied. We establish a one-to-one correspondence between vertices of degree at least 2d and nontrivial (excluding the trivial top eigenvalue) eigenvalues of A/ d outside of the asymptotic bulk [−2,2]. This correspondence implies that the transition characterized by the appearance of the eigenvalues outside of the asymptotic bulk takes place at the critical value d=d∗=1log4−1logN. For d<d∗, we obtain rigidity bounds on the locations of all eigenvalues outside the interval [−2,2], and for d>d∗, we show that no such eigenvalues exist. All of our estimates are quantitative with polynomial error probabilities. Our proof is based on a tridiagonal representation of the adjacency matrix and on a detailed analysis of the geometry of the neighbourhood of the large degree vertices. An important ingredient in our estimates is a matrix inequality obtained via the associated nonbacktracking matrix and an Ihara–Bass formula (Ann. Inst. Henri Poincaré Probab. Stat. 56 (2020) 2141–2161). Our argument also applies to sparse Wigner matrices, defined as the Hadamard product of A and a Wigner matrix, in which case the role of the degrees is replaced by the squares of the ℓ2-norms of the rows.

Parallel Newton–Chebyshev polynomial preconditioners for the conjugate gradient method

In this note, we exploit polynomial preconditioners for the conjugate gradient method to solve large symmetric positive definite linear systems in a parallel environment. We put in connection a specialized Newton method to solve the matrix equation X−1 = A and the Chebyshev polynomials for preconditioning. We propose a simple modification of one parameter which avoids clustering of extremal eigenvalues in order to speed-up convergence. We provide results on very large matrices (up to 8.6 billion unknowns) in a parallel environment showing the efficiency of the proposed class of preconditioners.

Limiting laws for extreme eigenvalues of large-dimensional spiked Fisher matrices with a divergent number of spikes

Consider the p×p matrix that is the product of a population covariance matrix and the inverse of another population covariance matrix. Suppose that their difference has a divergent rank with respect to p, when two samples of sizes n and T from the two populations are available, we construct its corresponding sample version. In the high-dimensional regime where both n and T are proportional to p, we investigate the limiting laws for extreme (spiked) eigenvalues of the sample (spiked) Fisher matrix when the number of spikes is divergent and these spikes are unbounded. We derive the convergence in probability of these spiked eigenvalues after scaling, and the central limit theorem for normalized spiked eigenvalues.

Extremal eigenvalues of the Conformal Laplacian under Sire–Xu normalization

Let (Mn,g) be a closed Riemannian manifold of dimension n≥3. We study the variational properties of the kth eigenvalue functional g̃∈[g]↦λk(Lg̃) under a non-volume normalization proposed by Sire–Xu. We discuss necessary conditions for the existence of extremal eigenvalues under such normalization. Also, we discuss the general existence problem when k=1.

Gaussian unitary ensemble with jump discontinuities and the coupled Painlevé II and IV systems

We study the orthogonal polynomials and the Hankel determinants associated with the Gaussian weight with two jump discontinuities. When the degree n is finite, the orthogonal polynomials and the Hankel determinants are shown to be connected with the coupled Painlevé IV system. Using this connection, we obtain a sequence of special function solutions to the coupled Painlevé IV system. In the double scaling limit as the jump discontinuities tend to the edge of the spectrum and the degree n grows to infinity, we establish the asymptotic expansions for the Hankel determinants and the orthogonal polynomials, which are expressed in terms of solutions of the coupled Painlevé II system. As applications, we re-derive the recently found Tracy–Widom type expressions for the gap probability of there being no eigenvalues in a finite interval near the extreme eigenvalue of large Hermitian matrix from the Gaussian unitary ensemble (GUE) and the limiting conditional distribution of the largest eigenvalue in the GUE by considering a thinned process.

Bounds of M-eigenvalues and strong ellipticity conditions for elasticity tensors

ABSTRACT Novel upper and lower bounds are derived for the M-eigenvalues of a fourth-order elasticity tensor by using the extreme eigenvalues of the symmetric matrices extracted from the given elasticity tensor. The new upper bound of the M-spectral radius is tighter than that given by S. Li et al. in [J Comput Appl Math. 2019;356]. Based on the new lower bound of the M-eigenvalues, a stronger sufficient condition is presented for the strong ellipticity of an elasticity tensor. Numerical examples are also given to demonstrate the results.

Convergence of Eigenvector Continuation.

Eigenvector continuation is a computational method that finds the extremal eigenvalues and eigenvectors of a Hamiltonian matrix with one or more control parameters. It does this by projection onto a subspace of eigenvectors corresponding to selected training values of the control parameters. The method has proven to be very efficient and accurate for interpolating and extrapolating eigenvectors. However, almost nothing is known about how the method converges, and its rapid convergence properties have remained mysterious. In this Letter, we present the first study of the convergence of eigenvector continuation. In order to perform the mathematical analysis, we introduce a new variant of eigenvector continuation that we call vector continuation. We first prove that eigenvector continuation and vector continuation have identical convergence properties and then analyze the convergence of vector continuation. Our analysis shows that, in general, eigenvector continuation converges more rapidly than perturbation theory. The faster convergence is achieved by eliminating a phenomenon that we call differential folding, the interference between nonorthogonal vectors appearing at different orders in perturbation theory. From our analysis we can predict how eigenvector continuation converges both inside and outside the radius of convergence of perturbation theory. While eigenvector continuation is a nonperturbative method, we show that its rate of convergence can be deduced from power series expansions of the eigenvectors. Our results also yield new insights into the nature of divergences in perturbation theory.

Fluctuations of extreme eigenvalues of sparse Erd\u0151s\u2013R\xe9nyi graphs

We consider a class of sparse random matrices which includes the adjacency matrix of the Erdős–Rényi graph {{mathcal {G}}}(N,p). We show that if N^{varepsilon } leqslant Np leqslant N^{1/3-varepsilon } then all nontrivial eigenvalues away from 0 have asymptotically Gaussian fluctuations. These fluctuations are governed by a single random variable, which has the interpretation of the total degree of the graph. This extends the result (Huang et al. in Ann Prob 48:916–962, 2020) on the fluctuations of the extreme eigenvalues from Np geqslant N^{2/9 + varepsilon } down to the optimal scale Np geqslant N^{varepsilon }. The main technical achievement of our proof is a rigidity bound of accuracy N^{-1/2-varepsilon } (Np)^{-1/2} for the extreme eigenvalues, which avoids the (Np)^{-1}-expansions from Erdős et al. (Ann Prob 41:2279–2375, 2013), Huang et al. (2020) and Lee and Schnelli (Prob Theor Rel Fields 171:543–616, 2018). Our result is the last missing piece, added to Erdős et al. (Commun Math Phys 314:587–640, 2012), He (Bulk eigenvalue fluctuations of sparse random matrices. arXiv:1904.07140), Huang et al. (2020) and Lee and Schnelli (2018), of a complete description of the eigenvalue fluctuations of sparse random matrices for Np geqslant N^{varepsilon }.