Eigenvalues Of Large Random Matrices Research Articles

One-level density for holomorphic cusp forms of arbitrary level

In 2000 Iwaniec, Luo, and Sarnak proved for certain families of L-functions associated to holomorphic newforms of square-free level that, under the Generalized Riemann Hypothesis, as the conductors tend to infinity the one-level density of their zeros matches the one-level density of eigenvalues of large random matrices from certain classical compact groups in the appropriate scaling limit. We remove the square-free restriction by obtaining a trace formula for arbitrary level by using a basis developed by Blomer and Milićević, which is of use for other problems as well.

On the spectral unfolding of chaotic and mixed systems

Random matrix theory (RMT) provides a framework to study the spectral fluctuations in physical systems. RMT is capable of making predictions for the fluctuations only after the removal of the secular properties of the spectrum. Spectral unfolding procedure is used to separate the local level fluctuations from overall energy dependence of the level separation. The unfolding procedure is not unique. Several studies showed that statistics of long-term correlation in the spectrum are very sensitive to the choice of the unfolding function in polynomial unfolding. This can give misleading results regarding the chaoticity of quantum systems. In this letter, we consider the spectra of ordered eigenvalues of large random matrices. We show that the main cause behind the reported sensitivity to the unfolding polynomial degree is the inclusion of specific extreme eigenvalue(s) in the unfolding process.

Topology and the Kardar–Parisi–Zhang universality class

We study the role of the topology of the background space on the one-dimensional Kardar–Parisi–Zhang (KPZ) universality class. To do so, we study the growth of balls on disordered 2D manifolds with random Riemannian metrics, generated by introducing random perturbations to a base manifold. As base manifolds we consider cones of different aperture angles θ, including the limiting cases of a cylinder (, which corresponds to an interface with periodic boundary conditions) and a plane (, which corresponds to an interface with circular geometry). We obtain that in the former case the radial fluctuations of the ball boundaries approach the Tracy–Widom (TW) distribution of the largest eigenvalue of random matrices in the Gaussian orthogonal ensemble (TW-GOE), while on cones with any aperture angle fluctuations correspond to the TW-GUE distribution related with the Gaussian unitary ensemble. We provide a topological argument to justify the relevance of TW-GUE statistics for cones, and state a conjecture which relates the KPZ universality subclass with the background topology.

Random geometry and the Kardar–Parisi–Zhang universality class

We consider a model of a quenched disordered geometry in which a random metric is defined on , which is flat on average and presents short-range correlations. We focus on the statistical properties of balls and geodesics, i.e., circles and straight lines. We show numerically that the roughness of a ball of radius R scales as , with a fluctuation exponent , while the lateral spread of the minimizing geodesic between two points at a distance L grows as , with wandering exponent value . Results on related first-passage percolation problems lead us to postulate that the statistics of balls in these random metrics belong to the Kardar–Parisi–Zhang universality class of surface kinetic roughening, with ξ and χ relating to critical exponents characterizing a corresponding interface growth process. Moreover, we check that the one-point and two-point correlators converge to the behavior expected for the Airy-2 process characterized by the Tracy–Widom (TW) probability distribution function of the largest eigenvalue of large random matrices in the Gaussian unitary ensemble (GUE). Nevertheless extreme-value statistics of ball coordinates are given by the TW distribution associated with random matrices in the Gaussian orthogonal ensemble. Furthermore, we also find TW–GUE statistics with good accuracy in arrival times.

Distributions for the eigenvalues of large random matrices generated from four manifolds

In this paper we will first study the spectrum of certain large Euclidean random matrices. The entries of these matrices are functions of n random points lp-norm uniformly sampled in N dimensional lp ball or lp sphere. Under the setting n/N → 0, as N and n tend to ∞, it is shown that the empirical distributions of eigenvalues of normalized Euclidean random matrices generated from the above two manifolds both converge weakly to semicircle law. We also investigate certain ‘sample-covariance’ type matrices whose entries are determined by n random positions from an N dimensional lp ellipsoid or its surface. When N → ∞ and n → ∞ with N/n → 0, we prove that their empirical spectral distribution converge to the same fixed distribution.

Transport through modes in random media

Excitations in complex media are superpositions of eigenstates that are referred to as 'levels' for quantum systems and 'modes' for classical waves. Although the Hamiltonian of a complex system may not be known or solvable, Wigner conjectured that the statistics of energy level spacings would be the same as for the eigenvalues of large random matrices. This has explained key characteristics of neutron scattering spectra. Subsequently, Thouless and co-workers argued that the metal-insulator transition in disordered systems could be described by a single parameter, the ratio of the average width and spacing of electronic energy levels: when this dimensionless ratio falls below unity, conductivity is suppressed by Anderson localization of the electronic wavefunction. However, because of spectral congestion due to the overlap of modes, even for localized waves, a comprehensive modal description of wave propagation has not been realized. Here we show that the field speckle pattern of transmitted radiation--in this case, a microwave field transmitted through randomly packed alumina spheres--can be decomposed into a sum of the patterns of the individual modes of the medium and the central frequency and linewidth of each mode can be found. We find strong correlation between modal field speckle patterns, which leads to destructive interference between modes. This allows us to explain complexities of steady state and pulsed transmission of localized waves and to harmonize wave and particle descriptions of diffusion.

Multicanonical sampling of rare events in random matrices

A method based on multicanonical Monte Carlo is applied to the calculation of large deviations in the largest eigenvalue of random matrices. The method is successfully tested with the Gaussian orthogonal ensemble, sparse random matrices, and matrices whose components are subject to uniform density. Specifically, the probability that all eigenvalues of a matrix are negative is estimated in these cases down to the values of ∼10(-200), a region where simple random sampling is ineffective. The method can be applied to any ensemble of matrices and used for sampling rare events characterized by any statistics.

Universal Fluctuations of Growing Interfaces: Evidence in Turbulent Liquid Crystals

We investigate growing interfaces of topological-defect turbulence in the electroconvection of nematic liquid crystals. The interfaces exhibit self-affine roughening characterized by both spatial and temporal scaling laws of the Kardar-Parisi-Zhang theory in 1+1 dimensions. Moreover, we reveal that the distribution and the two-point correlation of the interface fluctuations are universal ones governed by the largest eigenvalue of random matrices. This provides quantitative experimental evidence of the universality prescribing detailed information of scale-invariant fluctuations.

The First Two Eigenvalues of Large Random Matrices and Brody's Hypothesis on the Stability of Large Input–Output Systems

Bródy (1997) notices that for large random Leontief matrices, namely non-negative square matrices with all entries i.i.d., the ratio between the subdominant eigenvalue (in modulus) and the dominant eigenvalue declines generically to zero at a speed of the square root of the size of the matrix as the matrix size goes to infinity. Since then, several studies have been published in this journal in attempting to rigorously verify Bródy's conjecture. This short article, drawing upon some theorems obtained in recent years in the literature on empirical spectral distribution of random matrices, offers a short proof of Bródy's conjecture, and discusses briefly some related issues.

Universal Distribution Functions in Two-Dimensional Localized Systems

We find the conductance distribution function of the two-dimensional Anderson model in the strongly localized limit. The fluctuations of lng grow with lateral size as L1/3 and follow a universal distribution that depends on the type of leads. For narrow leads, it is the Tracy-Widom distribution, which appears in the problem of the largest eigenvalue of random matrices from the Gaussian unitary ensemble and in many other problems like the longest increasing subsequence of a permutation, directed polymers, or polynuclear growth. We also show that for wide leads the conductance follows a related, but different, distribution.

On the spacing distribution of the Riemann zeros: corrections to the asymptotic result

It has been conjectured that the statistical properties of zeros of the Riemann zeta function near $z = 1/2 + \ui E$ tend, as $E \to \infty$, to the distribution of eigenvalues of large random matrices from the Unitary Ensemble. At finite $E$ numerical results show that the nearest-neighbour spacing distribution presents deviations with respect to the conjectured asymptotic form. We give here arguments indicating that to leading order these deviations are the same as those of unitary random matrices of finite dimension $N_{\rm eff}=\log(E/2\pi)/\sqrt{12 \Lambda}$, where $\Lambda=1.57314 ...$ is a well defined constant.

Characteristic Polynomials of Random Matrices

Number theorists have studied extensively the connections between the distribution of zeros of the Riemann $\zeta$-function, and of some generalizations, with the statistics of the eigenvalues of large random matrices. It is interesting to compare the average moments of these functions in an interval to their counterpart in random matrices, which are the expectation values of the characteristic polynomials of the matrix. It turns out that these expectation values are quite interesting. For instance, the moments of order 2K scale, for unitary invariant ensembles, as the density of eigenvalues raised to the power $K^2$ ; the prefactor turns out to be a universal number, i.e. it is independent of the specific probability distribution. An equivalent behaviour and prefactor had been found, as a conjecture, within number theory. The moments of the characteristic determinants of random matrices are computed here as limits, at coinciding points, of multi-point correlators of determinants. These correlators are in fact universal in Dyson's scaling limit in which the difference between the points goes to zero, the size of the matrix goes to infinity, and their product remains finite.

Correlation functions of eigenvalues of multi-matrix models, and the limit of a time-dependent matrix

The universality of correlation functions of eigenvalues of large random matrices has been observed in various physical systems, and proved in some particular cases, as the Hermitian one-matrix model with polynomial potential. Here, we consider the more difficult case of a unidimensional chain of Hermitian matrices with first-neighbour couplings and polynomial potentials. An asymptotic expression of the orthogonal polynomials and a generalization of the Darboux-Christoffel theorem allow us to find new results for the correlations of eigenvalues of different matrices of the chain. Eventually, we consider the limit of the infinite chain of matrices, which can be interpreted as a time-dependent Hermitian one-matrix model, and give the correlation functions of eigenvalues at different times.

Correlations between eigenvalues of large random matrices with independent entries.

We derive the connected correlation functions for eigenvalues of large Hermitian random matrices with independently distributed elements using both a diagrammatic and a renormalization group (RG) inspired approach. With the diagrammatic method we obtain a general form for the one, two and three-point connected Green function for this class of ensembles when matrix elements are identically distributed, and then discuss the derivation of higher order functions by the same approach. Using the RG approach we re-derive the one and two-point Green functions and show they are unchanged by choosing certain ensembles with non-identically distributed elements. Throughout, we compare the Green functions we obtain to those from the class of ensembles with unitary invariant distributions and discuss universality in both ensemble classes.

Universality of the correlations between eigenvalues of large random matrices

The distribution of eigenvalues of random matrices appears in a number of physical situations, and it has been noticed that the resulting properties are universal, i.e. independent of specific details. Standard examples are provided by the universality of the conductance fluctuations from sample to sample in mesoscopic electronic systems, and by the spectrum of energy levels of a non-integrable classical hamiltonian (the so-called quantum chaos). The correlations between eigenvalues, measured on the appropriate scale, are described in all those cases by simple gaussian statistics. Similarly numerical experiments have revealed the universality of these correlations with respect to the probability measure of the random matrices. A simple renormalization group argument leads to a direct understanding of this universality; it is a consequence of the attractive nature of a gaussian fixed point. Detailed calculations of these correlations are given for a general probability distribution (in which the logarithm of the probability is the trace of a polynomial of the matrix); the universality is shown to follow from an explicit asymptotic form of the orthogonal polynomials with respect to a non-gaussian measure. In addition it is found that the connected correlations, when suitably smoothed, exhibit, even when the eigenvalues are not in the scaling region, a higher level of universality than the density of states.

Time scale of thermally activated diffusion in random systems: A new law of thermal activation.

Exact time scales are obtained for a class of stochastic models of thermally activated diffusion in random systems with a one-dimensional quantum reaction coordinate. The results are valid at any temperature. At low temperatures various forms of the thermal activation law of Arrhenius emerge, the dominant time scale of the diffusion depending on the temperature T according to ${T}^{p}$ exp(A/${k}_{B}$T), where A is the activation energy and p an exponent depending on the details of the models. These correspond to different bounded energy spectra for the randomly selected energy levels along the reaction coordinate. As an example we consider the Wigner semicircular distribution of eigenvalues of large random matrices. Another example considered is a Gaussian distribution of energy levels, and in this case a new thermal-activation law is obtained, having the form exp[${B}^{2}$/(${k}_{B}$T${)}^{2}$], where B is proportional to the width of the distribution. This form has recently been found in Monte Carlo experiments on large Ising-spin-glass models by Young.