Dynamical correlations for circular ensembles of random matrices

Circular Brownian motion models of random matrices were introduced by Dyson and describe the parametric eigenparameter correlations of unitary random matrices. For symmetric unitary, self-dual quaternion unitary and an analogue of antisymmetric Hermitian matrix initial conditions, Brownian dynamics toward the unitary symmetry is analyzed. The dynamical correlation functions of arbitrary number of Brownian particles at arbitrary number of times are shown to be written in the forms of quaternion determinants, similarly as in the case of Hermitian random matrix models.     

Universality in Orthogonal and Symplectic Invariant Matrix Models with Quartic Potential

In this work, we develop an orthogonal-polynomials approach for random matrices with orthogonal or symplectic invariant laws, called one-matrix models with polynomial potential in theoretical physics, which are a generalization of Gaussian random matrices. The representation of the correlation functions in these matrix models, via the technique of quaternion determinants, makes use of matrix kernels. We get new formulas for matrix kernels, generalizing the known formulas for Gaussian random matrices, which essentially express them in terms of the reproducing kernel of the theory of orthogonal polynomials. Finally, these formulas allow us to prove the universality of the local statistics of eigenvalues, both in the bulk and at the edge of the spectrum, for matrix models with two-band quartic potential by using the asymptotics given by Bleher and Its for the corresponding orthogonal polynomials.     

On the Partial Connection Between Random Matrices and Interacting Particle Systems

In the last decade there has been increasing interest in the fields of random matrices, interacting particle systems, stochastic growth models, and the connections between these areas. For instance, several objects appearing in the limit of large matrices arise also in the long time limit for interacting particles and growth models. Examples of these are the famous Tracy-Widom distribution functions and the Airy₂ process.     

Correlation functions for multi-matrix models and quaternion determinants

Using a close relationship to the Brownian motion model of random matrices, multi-matrix models in quantum field theory are analyzed. In the case many hermitian matrices are connected in a chain and one of them has a restricted (real symmetric, self dual real quaternion or antisymmetric hermitian) symmetry, the multi-matrix multi-level correlation functions are shown to have quaternion determinant forms.     

Pfaffian Expressions for Random Matrix Correlation Functions

It is well known that Pfaffian formulas for eigenvalue correlations are useful in the analysis of real and quaternion random matrices. Moreover the parametric correlations in the crossover to complex random matrices are evaluated in the forms of Pfaffians. In this article, we review the formulations and applications of Pfaffian formulas. For that purpose, we first present the general Pfaffian expressions in terms of the corresponding skew orthogonal polynomials. Then we clarify the relation to Eynard and Mehta’s determinant formula for hermitian matrix models and explain how the evaluation is simplified in the cases related to the classical orthogonal polynomials. Applications of Pfaffian formulas to random matrix theory and other fields are also mentioned.     

Random matrices as models for the statistics of quantum mechanics

Random matrices from the Gaussian unitary ensemble generate in a natural way unitary groups of evolution in finite-dimensional spaces. The statistical properties of this time evolution can be investigated by studying the time autocorrelation functions of dynamical variables. We prove general results on the decay properties of such autocorrelation functions in the limit of infinite-dimensional matrices. We discuss the relevance of random matrices as models for the dynamics of quantum systems that are chaotic in the classical limit.     

Superbosonization Formula and its Application to Random Matrix Theory

Starting from Gaussian random matrix models we derive a new supermatrix field theory model. In contrast to the conventional non-linear sigma models, the new model is applicable for any range of correlations of the elements of the random matrices. We clarify the domain of integration for the supermatrices, and give a demonstration of how the model works by calculating the density of states for an ensemble of almost diagonal matrices. It is also shown how one can reduce the supermatrix model to the conventional sigma model.     

Symmetrized Models of Last Passage Percolation and Non-Intersecting Lattice Paths

It has been shown that the last passage time in certain symmetrized models of directed percolation can be written in terms of averages over random matrices from the classical groups U(l), Sp(2l) and O(l). We present a theory of such results based on non-intersecting lattice paths, and integration techniques familiar from the theory of random matrices. Detailed derivations of probabilities relating to two further symmetrizations are also given.     

A metric measure for weight matrices of variable lengths—with applications to clustering and classification of hidden Markov models

We construct a metric measure among weight matrices that are commonly used in non-interacting statistical physics systems, computational biology problems, as well as in general applications such as hidden Markov models. The metric distance between two weight matrices is obtained via aligning the matrices and thus can be evaluated by dynamic programming. Capable of allowing reverse complements in distance evaluation, this metric accommodates both gapless and gapped alignments between two weight matrices. The distance statistics among random motifs is also studied. We find that the average square distance and its standard error grow with different powers of motif length, and the normalized square distance follows a Gaussian distribution for large motif lengths.     

Lyapunov exponents of large,sparse random matrices and the problem of directed polymers with complex random weights

We present results on two different problems: the Lyapunov exponent of large, sparse random matrices and the problem of polymers on a Cayley tree with random complex weights. We give an analytic expression for the largest Lyapunov exponent of products of random sparse matrices, with random elements located at random positions in the matrix. This expression is obtained through an analogy with the problem of random directed polymers on a Cayley tree (i.e., in the mean field limit), which itself can be solved using its relationship with random energy models (REM and GREM). For the random polymer problem with complex weights we find that, in addition to the high- and the low-temperature phases which were already known in the case of positive weights, the mean field theory predicts a new phase (phase III) which is dominated by interference effects.     

Random matrices applied to superconductivity in nano-particles

In this paper we introduce an approach in which the random matrices are applied to superconducting nano-particles, and obtain the effects of enhancement and attenuation simultaneously. We also explore the influence of magnetic fields on the superconductivity and the condensation energies in nano-particles. Comparisons with other models and some experimental results are given.     

Normal random matrix ensemble as a growth problem

In general or normal random matrix ensembles, the support of eigenvalues of large size matrices is a planar domain (or several domains) with a sharp boundary. This domain evolves under a change of parameters of the potential and of the size of matrices. The boundary of the support of eigenvalues is a real section of a complex curve. Algebro-geometrical properties of this curve encode physical properties of random matrix ensembles. This curve can be treated as a limit of a spectral curve which is canonically defined for models of finite matrices. We interpret the evolution of the eigenvalue distribution as a growth problem, and describe the growth in terms of evolution of the spectral curve. We discuss algebro-geometrical properties of the spectral curve and describe the wave functions (normalized characteristic polynomials) in terms of differentials on the curve. General formulae and emergence of the spectral curve are illustrated by three meaningful examples.     

Lattices of matrices

We study a new class of matrix models, formulated on a lattice. On each site are N states with random energies governed by a gaussian random matrix hamiltonian. The states on different sites are coupled randomly. We calculate the density of and correlation between the eigenvalues of the total hamiltonian in the large-N limit. We find that this correlation exhibits the same type of universal behavior we discovered recently. Several derivations of this result are given. This class of random matrices allows us to model the transition between the “localized” and “extended” regimes within the limited context of random matrix theory.     

Non-hermitian random matrix models

We introduce an extension of the diagrammatic rules in random matrix theory and apply it to non-hermitian random matrix models using the 1/N approximation. A number of one-and two-point functions are evaluated on their holomorphic and non-holomorphic supports to leading order in 1/N. The one-point functions describe the distribution of eigenvalues, while the two-point functions characterize their macroscopic cotrelations. The generic form for the two-point functions is obtained, generalizing the concept of macroscopic universality to non-hermitian random matrices. We show that the holomorphic and non-holomorphic one- and two-point functions condition the behavior of pertinent partition functions to order O(1/N). We derive explicit conditions for the location and distribution of their singularities. Most of our analytical results are found to be in good agreement with numerical calculations using large ensembles of complex matrices.     

Entanglement generation of nearly random operators

We study the entanglement generation of operators whose statistical properties approach those of random matrices but are restricted in some way. These include interpolating ensemble matrices, where the interval of the independent random parameters are restricted, pseudorandom operators, where there are far fewer random parameters than required for random matrices, and quantum chaotic evolution. Restricting randomness in different ways allows us to probe connections between entanglement and randomness. We comment on which properties affect entanglement generation and discuss ways of efficiently producing random states on a quantum computer.     

Replica field theories,painlevé transcendents,and exact correlation functions

Exact solvability is claimed for nonlinear replica sigma models derived in the context of random matrix theories. Contrary to other approaches reported in the literature, the framework outlined does not rely on traditional "replica symmetry breaking" but rests on a previously unnoticed exact relation between replica partition functions and Painlevé transcendents. While expected to be applicable to matrix models of arbitrary symmetries, the method is used to treat fermionic replicas for the Gaussian unitary ensemble (GUE), chiral GUE (symmetry classes A and AIII in Cartan classification) and Ginibre's ensemble of complex non-Hermitian random matrices. Further applications are briefly discussed.     

Long-range correlations in time series generated by time-fractional diffusion: A numerical study

Time series models showing power law tails in autocorrelation functions are common in econometrics. A special non-Markovian model for such kind of time series is provided by the random walk introduced by Gorenflo et al. as a discretization of time fractional diffusion. The time series so obtained are analyzed here from a numerical point of view in terms of autocorrelations and covariance matrices.     

On the 1/n Expansion for Some Unitary Invariant Ensembles of Random Matrices

We present a version of the 1/n-expansion for random matrix ensembles known as matrix models. The case where the support of the density of states of an ensemble consists of one interval and the case where the density of states is even and its support consists of two symmetric intervals is treated. In these cases we construct the expansion scheme for the Jacobi matrix determining a large class of expectations of symmetric functions of eigenvalues of random matrices, prove the asymptotic character of the scheme and give an explicit form of the first two terms. This allows us, in particular, to clarify certain theoretical physics results on the variance of the normalized traces of the resolvent of random matrices. We also find the asymptotic form of several related objects, such as smoothed squares of certain orthogonal polynomials, the normalized trace and the matrix elements of the resolvent of the Jacobi matrices, etc. Received: 9 November 2000 / Accepted: 26 July 2001     

Precise toppling balance, quenched disorder, and universality for sandpiles

A single sandpile model with quenched random toppling matrices captures the crucial features of different models of self-organized criticality. With symmetric matrices avalanche statistics falls in the multiscaling Bak-Tang-Wiesenfeld universality class. In the asymmetric case the simple scaling of the Manna model is observed. The presence or absence of a precise toppling balance between the amount of sand released by a toppling site and the total quantity the same site receives when all its neighbors topple once determines the appropriate universality class.     

On Asymptotic Behavior of Multilinear Eigenvalue Statistics of Random Matrices

We prove the Law of Large Numbers and the Central Limit Theorem for analogs of U- and V- (von Mises) statistics of eigenvalues of random matrices as their size tends to infinity. We show first that for a certain class of test functions (kernels), determining the statistics, the validity of these limiting laws reduces to the validity of analogous facts for certain linear eigenvalue statistics. We then check the conditions of the reduction statements for several most known ensembles of random matrices. The reduction phenomenon is well known in statistics, dealing with i.i.d. random variables. It is of interest that an analogous phenomenon is also the case for random matrices, whose eigenvalues are strongly dependent even if the entries of matrices are independent.