Gaussian Ensemble Research Articles

Linear differential equations for the resolvents of the classical matrix ensembles

The spectral density for random matrix [Formula: see text] ensembles can be written in terms of the average of the absolute value of the characteristic polynomial raised to the power of [Formula: see text], which for even [Formula: see text] is a polynomial of degree [Formula: see text]. In the cases of the classical Gaussian, Laguerre, and Jacobi weights, we show that this polynomial, and moreover, the spectral density itself, can be characterized as the solution of a linear differential equation of degree [Formula: see text]. This equation, and its companion for the resolvent, are given explicitly for [Formula: see text] and [Formula: see text] for all three classical cases, and also for [Formula: see text] in the Gaussian case. Known dualities for the spectral moments relating [Formula: see text] to [Formula: see text] then imply corresponding differential equations in the case [Formula: see text], and for the Gaussian ensemble, the case [Formula: see text]. We apply the differential equations to give a systematic derivation of recurrences satisfied by the spectral moments and by the coefficients of their [Formula: see text] expansions, along with first-order differential equations for the coefficients of the [Formula: see text] expansions of the corresponding resolvents. We also present the form of the differential equations when scaled at the hard or soft edges.

A gaussian process-based iterative Ensemble Kalman Filter for parameter estimation of unsaturated flow

Water flow in the unsaturated zone is an important component of the water cycle. Accurate estimation of soil hydraulic parameters ensures precise simulations of water flow in the unsaturated zone. In this study, a Gaussian Process-based Iterative Ensemble Kalman Filter method (GPIEnKF) is proposed and applied for estimating soil hydraulic parameters for two-dimensional soil water flow. The method involves a surrogate model for two-dimensional soil water flow that is based on the Gaussian Process method. The accuracy and efficiency of the GPIEnKF method are validated using two synthetic cases and a real dataset involving a field experiment with drip irrigation. The impact of the layout of observation points as well as the number of training base points and observation points on the estimation of parameters are also analyzed. The results show that the surrogate model can accurately predict pressure heads at observation points. The layout of observation points that precisely describes infiltration water movement allows the surrogate model to better emulate the original model, thereby improving the accuracy of parameter estimation. The number of training base points is found to have only a small influence on parameter estimation. The accuracy of parameter estimation and pressure head predictions can be further improved by increasing the number of observation points. However, the accuracy of predictions is affected by the uncertainty of boundary conditions and the soil spatial heterogeneity. The GPIEnKF method can effectively estimate multiple parameters characterizing water flow in layered soils. As compared with the standard Iterative Ensemble Kalman Filter, the GPIEnKF method can greatly improve computational efficiency while obtaining comparable results. The GPIEnKF method is an efficient tool for parameter estimation of multi-dimensional soil water flow.

Quantum metric statistics for random-matrix families

The quantum metric tensor G ij for parameterised families of quantum states, in particular the trace G = trG ij , depends on the symmetry of the system (e.g. time-reversal), and the dimension N of the underlying matrices. Modelling the families by the stationary Gaussian ensembles of random-matrix, theory, we calculate the probability distribution of G, exactly for N = 2, and approximately for N = 3 and N → ∞. Codimension arguments establish the scalings of the distributions near the singularities at G → ∞ and G = 0, near which asymptotics gives the explicit analytic behaviour. Numerical simulations support the theory.

Identical fixed points in state evolutions of AMP and VAMP

AMP and VAMP are two different algorithms that solve high-dimensional standard linear regression problems effectively and efficiently. In previous studies, they were long observed to have identical fixed points in their state evolutions (SEs), however, a formal justification is still missing. This issue is addressed in the paper. Instead of looking directly into the problem, we consider a scope more ambitious which puts aside all assumptions underlying the proven SEs. We then show that: 1) Having the spectrum of a Marchenko-Pastur law is a necessary and sufficient condition for the AMP-style and VAMP-style SEs to share an identical fixed point; 2) As the zero-mean i.i.d. Gaussian ensemble have a Marchenko-Pastur spectrum and satisfies all conditions required by the proven AMP SE and VAMP SE, this particular ensemble is the first example of having an identical fixed point. After that, we conjecture the proven VAMP SE might be extended to cover ensembles that break the right rotational invariance, among which the i.i.d. zero-mean sub-Gaussian ensemble could be a second example to yield identical fixed points. This conjecture is supported by experiments from wireless communications and compressive sampling.

Boundary Effect on the Nodal Length for Arithmetic Random Waves, and Spectral Semi-correlations

We test M. Berry’s ansatz on nodal deficiency in presence of boundary. The square billiard is studied, where the high spectral degeneracies allow for the introduction of a Gaussian ensemble of random Laplace eigenfunctions (“boundary-adapted arithmetic random waves”). As a result of a precise asymptotic analysis, two terms in the asymptotic expansion of the expected nodal length are derived, in the high energy limit along a generic sequence of energy levels. It is found that the precise nodal deficiency or surplus of the nodal length depends on arithmetic properties of the energy levels, in an explicit way. To obtain the said results we apply the Kac–Rice method for computing the expected nodal length of a Gaussian random field. Such an application uncovers major obstacles, e.g. the occurrence of “bad” subdomains, that, one hopes, contribute insignificantly to the nodal length. Fortunately, we were able to reduce this contribution to a number theoretic question of counting the “spectral semi-correlations”, a concept joining the likes of “spectral correlations” and “spectral quasi-correlations” in having impact on the nodal length for arithmetic dynamical systems. This work rests on several breakthrough techniques of J. Bourgain, whose interest in the subject helped shaping it to high extent, and whose fundamental work on spectral correlations, joint with E. Bombieri, has had a crucial impact on the field.

Distribution of the ratio of two consecutive level spacings in orthogonal to unitary crossover ensembles.

The ratio of two consecutive level spacings has emerged as a very useful metric in investigating universal features exhibited by complex spectra. It does not require the knowledge of density of states and is therefore quite convenient to compute in analyzing the spectrum of a general system. The Wigner-surmise-like results for the ratio distribution are known for the invariant classes of Gaussian random matrices. However, for the crossover ensembles, which are useful in modeling systems with partially broken symmetries, corresponding results have remained unavailable so far. In this work, we derive exact results for the distribution and average of the ratio of two consecutive level spacings in the Gaussian orthogonal to unitary crossover ensemble using a 3×3 random matrix model. This crossover is useful in modeling time-reversal symmetry breaking in quantum chaotic systems. Although based on a 3×3 matrix model, our results can also be applied in the study of large spectra, provided the symmetry-breaking parameter facilitating the crossover is suitably scaled. We substantiate this claim by considering Gaussian and Laguerre crossover ensembles comprising large matrices. Moreover, we apply our result to investigate the violation of time-reversal invariance in the quantum kicked rotor system.

Restriction of 3D arithmetic Laplace eigenfunctions to a plane

We consider a random Gaussian ensemble of Laplace eigenfunctions on the 3D torus, and investigate the 1-dimensional Hausdorff measure (‘length’) of nodal intersections against a smooth 2-dimensional toral sub-manifold (‘surface’). A prior result of ours prescribed the expected length, universally proportional to the area of the reference surface, times the wavenumber, independent of the geometry. In this paper, for surfaces contained in a plane, we give an upper bound for the nodal intersection length variance, depending on the arithmetic properties of the plane. The bound is established via estimates on the number of lattice points in specific regions of the sphere.

Eigenstate Thermalization and Rotational Invariance in Ergodic Quantum Systems.

Generic rotationally invariant matrix models satisfy a simple relation: the probability distribution of half the difference between any two diagonal elements and the one of off-diagonal elements are the same. In the spirit of the eigenstate thermalization hypothesis, we test the hypothesis that the same relation holds in quantum systems that are nonlocalized, when one considers small energy differences. The relation provides a stringent test of the eigenstate thermalization hypothesis beyond the Gaussian ensemble. We apply it to a disordered spin chain, the Sachdev-Ye-Kitaev model, and a Floquet system.

Continuum Goldstone spectrum of two-color QCD at finite density with staggered quarks

We carry out lattice simulations of two-color QCD and spectroscopy at finite density with two flavors of rooted-staggered quarks and a diquark source term. As in a previous four-flavor study, for small values of the inverse gauge coupling we observe a Goldstone spectrum which reflects the symmetry-breaking pattern of a Gaussian symplectic chiral random-matrix ensemble (GSE) with Dyson index $\beta_D=4$, which corresponds to any-color QCD with adjoint quarks in the continuum instead of QC$_2$D wih fundamental quarks. We show that this unphysical behavior occurs only inside of the bulk phase of $SU(2)$ gauge theory, where the density of $Z_2$ monopoles is high. Using an improved gauge action and a somewhat larger inverse coupling to suppress these monopoles, we demonstrate that the continuum Goldstone spectrum of two-color QCD, corresponding to a Gaussian orthogonal ensemble (GOE) with Dyson index $\beta_D=1$, is recovered also with rooted-staggered quarks once simulations are performed away from the bulk phase. We further demonstrate how this change of random-matrix ensemble is reflected in the distribution of eigenvalues of the Dirac operator. By computing the unfolded level spacings inside and outside of the bulk phase, we demonstrate that, starting with the low-lying eigenmodes which determine the infrared physics, the distribution of eigenmodes continuously changes from the GSE to the GOE one as monopoles are suppressed.

Higher-order pathwise theory of fluctuations in stochastic homogenization

We consider linear elliptic equations in divergence form with stationary random coefficients of integrable correlations. We characterize the fluctuations of a macroscopic observable of a solution to relative order $$\frac{d}{2}$$ , where d is the spatial dimension; the fluctuations turn out to be Gaussian. As for previous work on the leading order, this higher-order characterization relies on a pathwise proximity of the macroscopic fluctuations of a general solution to those of the (higher-order) correctors, via a (higher-order) two-scale expansion injected into the “homogenization commutator”, thus confirming the scope of this notion. This higher-order generalization sheds a clearer light on the algebraic structure of the higher-order versions of correctors, flux correctors, two-scale expansions, and homogenization commutators. It reveals that in the same way as this algebra provides a higher-order theory for microscopic spatial oscillations, it also provides a higher-order theory for macroscopic random fluctuations, although both phenomena are not directly related. We focus on the model framework of an underlying Gaussian ensemble, which allows for an efficient use of (second-order) Malliavin calculus for stochastic estimates. On the technical side, we introduce annealed Calderon–Zygmund estimates for the elliptic operator with random coefficients, which conveniently upgrade the known quenched large-scale estimates.

Linear statistics of random matrix ensembles at the spectrum edge associated with the Airy kernel

In this paper, we study the large N behavior of the moment-generating function (MGF) of the linear statistics of N×N Hermitian matrices in the Gaussian unitary, symplectic, orthogonal ensembles (GUE, GSE, GOE) and Laguerre unitary, symplectic, orthogonal ensembles (LUE, LSE, LOE) at the edge of the spectrum. From the finite N Fredholm determinant expression of the MGF of the linear statistics [35], we find the large N asymptotics of the MGF associated with the Airy kernel in these Gaussian and Laguerre ensembles. Then we obtain the mean and variance of the suitably scaled linear statistics. We show that there is an equivalence between the large N behavior of the MGF of the scaled linear statistics in Gaussian and Laguerre ensembles, which leads to the statistical equivalence between the mean and variance of suitably scaled linear statistics in Gaussian and Laguerre ensembles. In the end, we use the Coulomb fluid method to obtain the mean and variance of another type of linear statistics in GUE, which reproduces the result of Basor and Widom [7].

Predicting Convergence Rate of Namaklan Twin Tunnels Using Machine Learning Methods

Convergence prediction of tunnels has always been one of the important issues of geotechnical projects. Developing prediction models is a good approach to predict convergence, when there is no knowledge of the possibility of convergence occurrence in the future. In this study, convergence rates of two tunnels from the Namaklan twin tunnel were predicted by different types of machine learning methods. Artificial neural networks (ANNs), multivariate linear regression (MLR), multivariate nonlinear regression (MNR), support vector regression (SVR), Gaussian process regression (GPR), regression trees and ensembles of trees (ET) were applied to predict the convergence rate (CR). The six parameters of cohesion (c), internal friction angle (Φ), uniaxial compressive strength of rock mass (σc), rock mass rating, overburden height (H) and the number of installed rock bolts (NB) were selected as predictor parameters. The dataset was collected via field investigations and laboratory experiments. The results showed that the MLP–ANN model can successfully predict the CR with the determination coefficient (R2) of 0.93. The RBF–ANN model is also successful in predicting the CR with R2 of 0.81. The SVR, MNR and MLR models were also constructed to obtain an empirical formula for predicting the CR. Comparing among the three models showed that the SVR model is more successful (R2 = 0.66) than the MNR and MLR models with R2 of 0.65 and 0.61, respectively. However, the SVR model is placed in the next rank of the ANN models. Among the rest models, except the ET model (R2 = 0.66), the RT and GPR models have no good capability for the prediction of the CR. In total, assessing the statistical indices indicated that the ANNs are superior to the other models in predicting the CR. However, the SVR model could be considered to be a reliable predictive model for convergence rate estimation.

On the Error in Phase Transition Computations for Compressed Sensing

Evaluating the statistical dimension is a common tool to determine the asymptotic phase transition in compressed sensing problems with Gaussian ensemble. Unfortunately, the exact evaluation of the statistical dimension is very difficult and it has become standard to replace it with an upper-bound. To ensure that this technique is suitable, [1] has introduced an upper-bound on the gap between the statistical dimension and its approximation. In this work, we first show that the error bound in [1] in some low-dimensional models such as total variation and $\ell_1$ analysis minimization becomes poorly large. Next, we develop a new error bound which significantly improves the estimation gap compared to [1]. In particular, unlike the bound in [1] that is not applicable to settings with overcomplete dictionaries, our bound exhibits a decaying behavior in such cases.

Context Vectors are Reflections of Word Vectors in Half the Dimensions

This paper takes a step towards theoretical analysis of the relationship between word embeddings and context embeddings in models such as word2vec. We start from basic probabilistic assumptions on the nature of word vectors, context vectors, and text generation. These assumptions are supported either empirically or theoretically by the existing literature. Next, we show that under these assumptions the widely-used word-word PMI matrix is approximately a random symmetric Gaussian ensemble. This, in turn, implies that context vectors are reflections of word vectors in approximately half the dimensions. As a direct application of our result, we suggest a theoretically grounded way of tying weights in the SGNS model.

Universal broadening of zero modes: A general framework and identification.

We consider the smallest eigenvalues of perturbed Hermitian operators with zero modes, either topological or system specific. To leading order for small generic perturbation we show that the corresponding eigenvalues broaden to a Gaussian random matrix ensemble of size ν×ν, where ν is the number of zero modes. This observation unifies and extends a number of results within chiral random matrix theory and effective field theory and clarifies under which conditions they apply. The scaling of the former zero modes with the volume differs from the eigenvalues in the bulk, which we propose as an indicator to identify them in experiments. These results hold for all 10 symmetric spaces in the Altland-Zirnbauer classification and build on two facts. First, the broadened zero modes decouple from the bulk eigenvalues and, second, the mixing from eigenstates of the perturbation form a central limit theorem argument for matrices.

GSE spectra in uni-directional quantum systems

Generically, spectral statistics of spinless systems with time reversal invariance (TRI) and chaotic dynamics are well-described by the Gaussian orthogonal ensemble (GOE). However, if an additional symmetry is present, the spectrum can be split into independent sectors which statistics depend on the type of the group’s irreducible representation. In particular, this allows for the construction of TRI quantum graphs with spectral statistics characteristic of the Gaussian symplectic ensembles (GSE). To this end one usually has to use groups admitting pseudo-real irreducible representations. In this paper we show how GSE spectral statistics can be realized in TRI systems with simpler symmetry groups lacking pseudo-real representations. As an application, we provide a class of quantum graphs with only C4 rotational symmetry possessing GSE spectral statistics.

Level crossing in random matrices. II. Random perturbation of a random matrix

In this paper we study the distribution of level crossings for the spectra of linear families where A and B are square matrices independently chosen from some given Gaussian ensemble and is a complex-valued parameter. We formulate a number of theoretical and numerical results for the classical Gaussian ensembles and some of their generalisations. Besides, we present intriguing numerical information about the monodromy distribution in case of linear families for the classical Gaussian ensembles of -matrices.

Spectral statistics of non-Hermitian random matrix ensembles

Recently Burkhardt et al. introduced the [Formula: see text]-checkerboard random matrix ensembles, which have a split limiting behavior of the eigenvalues (in the limit all but [Formula: see text] of the eigenvalues are on the order of [Formula: see text] and converge to semi-circular behavior, with the remaining [Formula: see text] of size [Formula: see text] and converging to hollow Gaussian ensembles). We generalize their work to consider non-Hermitian ensembles with complex eigenvalues; instead of a blip new behavior is seen, ranging from multiple satellites to annular rings. These results are based on moment method techniques adapted to the complex plane as well as analysis of singular values.

Circular Kardar-Parisi-Zhang interfaces evolving out of the plane.

Circular KPZ interfaces spreading radially in the plane have Gaussian unitary ensemble (GUE) Tracy-Widom (TW) height distribution (HD) and Airy_{2} spatial covariance, but what are their statistics if they evolve on the surface of a different background space, such as a bowl, a mountain, or any surface of revolution? To give an answer to this, we report here extensive numerical analyses of several one-dimensional KPZ models on substrates whose size enlarges as 〈L(t)〉=L_{0}+ωt^{γ}, while their mean height 〈h〉 increases as usual [〈h〉∼t]. We show that the competition between the L enlargement and the correlation length (ξ≃ct^{1/z}) plays a key role in the asymptotic statistics of the interfaces. While systems with γ>1/z have HDs given by GUE and the interface width increasing as w∼t^{β}, for γ<1/z the HDs are Gaussian, in a correlated regime where w∼t^{αγ}. For the special case γ=1/z, a continuous class of distributions exists, which interpolate between Gaussian (for small ω/c) and GUE (for ω/c≫1). Interestingly, the HD seems to agree with the Gaussian symplectic ensemble (GSE) TW distribution for ω/c≈10. Despite the GUE HDs for γ>1/z, the spatial covariances present a strong dependence on the parameters ω and γ, agreeing with Airy_{2} only for ω≫1, for a given γ, or when γ=1, for a fixed ω. These results considerably generalize our knowledge on 1D KPZ systems, unveiling the importance of the background space on their statistics.

Singular vector distribution of sample covariance matrices

AbstractWe consider a class of sample covariance matrices of the formQ=TXX*T*, whereX= (xij) is anM×Nrectangular matrix consisting of independent and identically distributed entries, andTis a deterministic matrix such thatT*Tis diagonal. Assuming thatMis comparable toN, we prove that the distribution of the components of the right singular vectors close to the edge singular values agrees with that of Gaussian ensembles provided the first two moments ofxijcoincide with the Gaussian random variables. For the right singular vectors associated with the bulk singular values, the same conclusion holds if the first four moments ofxijmatch those of the Gaussian random variables. Similar results hold for the left singular vectors if we further assume thatTis diagonal.