finite time fault detection filter design for discrete time interconnected systems with average dwell time abstract in this paper we present an approach to finite time robust fault detection filter design for interconnected systems with disconnected interconnections among these subsystems taking various disconnected interconnections into account a switched scheme is proposed to analysis these complex cases by modeling these cases of disconnections into independent subsystems a switching signal is introduced to specify which subsystem is activated in the aid of the average dwell time method sufficient conditions on existence of the desired filter are derived in term of linear matrices inequalities which guarantee the augmented system is finite time boundedness and achieve the h performance a numerical example is used to illustrate the effectiveness of the proposed approach

statistics of epicenters in the olami feder christensen model in two and three dimensions abstract recently abe and suzuki pointed out that epicenters of earthquakes could be connected in order to generate a graph with properties of a scale free network we have shown that the olami feder christensen ofc model for the dynamics of earthquakes defined both in square and cubic lattices is able to reproduce this new behavior the distribution of distances between successive earthquakes is also presented our results indicate the robustness of the ofc model to describe earthquake dynamics surprisingly we found that only the non conservative version of the ofc model generates a network with scale free properties the conservative version instead behaves like a random graph the distribution of distances in 3 d and the distribution of connectivities in a 2 d lattice confirm the differences observed between the conservative and non conservative regime

eigenvalue statistics of random real matrices completing ginibre s work we determine the joint probability density of eigenvalues in a gaussian ensemble of real asymmetric matrices which is invariant under orthogonal transformations the symmetry parameter r may vary from 1 antisymmetric ensemble through 0 completely asymmetric ensemble to 1 symmetric ensemble the elliptic law for the average density of eigenvalues in the limit of large dimension is recovered matrices of the type considered appear in models for neural network dynamics and dissipative quantum dynamics

skew orthogonal polynomials and random matrix ensembles there is considerable interest in understanding the relation between random matrix ensembles and quantum chaotic systems in the context of the universality of energy level correlations in this connection while gaussian ensembles of random matrices have been studied extensively not much is known about ensembles with non gaussian weight functions dyson has shown that the n level correlation functions can be expressed in terms of a kernel function involving orthogonal and skew orthogonal polynomials orthogonal for matrix ensembles with unitary invariance and skew orthogonal for ensembles with orthogonal and symplectic invariances we have obtained the following results 1 skew orthogonal polynomials of both types are derived for the jacobi class of weight functions including the limiting cases of associated laguerre and hermite or gaussian 2 matrix integral representations are given for the general weight functions 3 asymptotic forms of the polynomials are obtained rigorously for the jacobi class and in the form of an ansatz for the general case 4 for the three types of ensembles the asymptotic n level correlation functions with appropriate scaling are shown to be universal being independent of the weight function and location in the spectrum and identical with the well known gaussian results this provides a rigorous justification for the universality of the gaussian ensemble results observed in quantum chaotic systems as expected the level density is not universal

derivation of determinantal structures for random matrix ensembles in a new way there are several methods to treat ensembles of random matrices in symmetric spaces circular matrices chiral matrices and others orthogonal polynomials and the supersymmetry method are particular powerful techniques here we present a new approach to calculate averages over ratios of characteristic polynomials at first sight paradoxically one can call our approach supersymmetry without supersymmetry because we use structures from supersymmetry without actually mapping onto superspaces we address two kinds of integrals which cover a wide range of applications for random matrix ensembles for probability densities factorizing in the eigenvalues we find determinantal structures in a unifying way as a new application we derive an expression for the k point correlation function of an arbitrary rotation invariant probability density over the hermitian matrices in the presence of an external field

robustness of scale invariance in models with self organized criticality a random neighbor extremal stick slip model is introduced in the thermodynamic limit the distribution of states has a simple analytical form and the mean avalanche size as a function of the coupling parameter is exactly calculable the system is critical only at a special point j c in coupling parameter space however the critical region around this point where approximate scale invariance holds is very large suggesting a mechanism for explaining the ubiquity of power laws in nature

mixing of orthogonal and skew orthogonal polynomials and its relation to wilson rmt the unitary wilson random matrix theory is an interpolation between the chiral gaussian unitary ensemble and the gaussian unitary ensemble this new way of interpolation is also reflected in the orthogonal polynomials corresponding to such a random matrix ensemble although the chiral gaussian unitary ensemble and the gaussian unitary ensemble are associated with the dyson index 2 the intermediate ensembles exhibit a mixing of orthogonal polynomials and skew orthogonal polynomials we consider the hermitian and the non hermitian wilson random matrix and derive the corresponding polynomials their recursion relations christoffel darboux like formulas rodrigues formulas and representations as random matrix averages in a unifying way with the help of these results we derive the unquenched k point correlation function of the hermitian and the non hermitian wilson random matrix in terms of two flavor partition functions only this representation is due to a pfaffian factorization it drastically simplifies the expressions which can be easily numerically evaluated it also serves as a good starting point for studying the wilson dirac operator in the regime of lattice quantum chromodynamics

ratios of characteristic polynomials in complex matrix models we compute correlation functions of inverse powers and ratios of characteristic polynomials for random matrix models with complex eigenvalues compact expressions are given in terms of orthogonal polynomials in the complex plane as well as their cauchy transforms generalizing previous expressions for real eigenvalues we restrict ourselves to ratios of characteristic polynomials over their complex conjugate

skew orthogonal laguerre polynomials for chiral real asymmetric random matrices we apply the method of skew orthogonal polynomials sop in the complex plane to asymmetric random matrices with real elements belonging to two different classes explicit integral representations valid for arbitrary weight functions are derived for the sop and for their cauchy transforms given as the expectation values of traces and determinants or their inverses respectively our proof uses the fact that the joint probability distribution function for all combinations of real eigenvalues and complex conjugate eigenvalue pairs can be written as a product examples for the sop are given in terms of laguerre polynomials for the chiral ensemble also called the non hermitian real wishart laguerre ensemble both without and with the insertion of characteristic polynomials such characteristic polynomials play the role of mass terms in applications to complex dirac spectra in field theory in addition for the elliptic real ginibre ensemble we recover the sop of forrester and nagao in terms of hermite polynomials

real eigenvalue statistics for products of asymmetric real gaussian matrices abstract random matrices formed from i i d standard real gaussian entries have the feature that the expected number of real eigenvalues is non zero this property persists for products of such matrices independently chosen and moreover it is known that as the number of matrices in the product tends to infinity the probability that all eigenvalues are real tends to unity we quantify the distribution of the number of real eigenvalues for products of finite size real gaussian matrices by giving an explicit pfaffian formula for the probability that there are exactly k real eigenvalues as a determinant with entries involving particular meijer g functions we also compute the explicit form of the pfaffian correlation kernel for the correlation between real eigenvalues and the correlation between complex eigenvalues the simplest example of these the eigenvalue density of the real eigenvalues gives by integration the expected number of real eigenvalues our ability to perform these calculations relies on the construction of certain skew orthogonal polynomials in the complex plane the computation of which is carried out using their relationship to particular random matrix averages

surprising pfaffian factorizations in random matrix theory with dyson index β 2 in recent decades determinants and pfaffians were found for eigenvalue correlations of various random matrix ensembles these structures simplify the average over a large number of ratios of characteristic polynomials to integrations over one and two characteristic polynomials only up to now it was thought that determinants occur for ensembles with dyson index 2 whereas pfaffians only for ensembles with 1 4 we derive a non trivial pfaffian determinant for 2 random matrix ensembles which is similar to the one for 1 4 thus it unveils a hidden universality of this structure we also give a general relation between the orthogonal polynomials related to the determinantal structure and the skew orthogonal polynomials corresponding to the pfaffian as a particular example we consider the chiral unitary ensembles in great detail

general eigenvalue correlations for the real ginibre ensemble we rederive in a simplified version the lehmann sommers eigenvalue distribution for the gaussian ensemble of asymmetric real matrices invariant under real orthogonal transformations as a basis for a detailed derivation of a pfaffian generating functional for n point densities this produces a simple free fermion diagram expansion for the correlations leading to quaternion determinants in each order n all will explicitly be given with the help of a very simple symplectic kernel for even dimension n the kernel is valid for both complex and real eigenvalues and describes a deep connection between both a slight modification by an artificial additional grassmannian also solves the more complicated odd n case as illustration we present some numerical results in the space of complex eigenvalue n tuples

eigenvalue statistics of the real ginibre ensemble the real ginibre ensemble consists of random n ifmmode times else texttimes fi n matrices formed from independent and identically distributed standard gaussian entries by using the method of skew orthogonal polynomials the general n point correlations for the real eigenvalues and for the complex eigenvalues are given as n ifmmode times else texttimes fi n pfaffians with explicit entries a computationally tractable formula for the cumulative probability density of the largest real eigenvalue is presented this is relevant to may s stability analysis of biological webs

asymptotics of skew orthogonal polynomials exact integral expressions of the skew orthogonal polynomials involved in orthogonal 1 and symplectic 4 random matrix ensembles are obtained the even rank skew orthogonal polynomials are average characteristic polynomials of random matrices from there asymptotics of the skew orthogonal polynomials are derived

asymptotic coincidence of the statistics for degenerate and non degenerate correlated real wishart ensembles the correlated wishart model provides the standard benchmark when analyzing time series of any kind unfortunately the real case which is the most relevant one in applications poses serious challenges for analytical calculations often these challenges are due to square root singularities which cannot be handled using common random matrix techniques we present a new way to tackle this issue using supersymmetry we carry out an anlaytical study which we support by numerical simulations for large but finite matrix dimensions we show that statistical properties of the fully correlated real wishart model generically approach those of a correlated real wishart model with doubled matrix dimensions and doubly degenerate empirical eigenvalues this holds for the local and global spectral statistics with monte carlo simulations we show that this is even approximately true for small matrix dimensions we explicitly investigate the k point correlation function as well as the distribution of the largest eigenvalue for which we find a surprisingly compact formula in the doubly degenerate case moreover we show that on the local scale the k point correlation function exhibits the sine and the airy kernel in the bulk and at the soft edges respectively we also address the positions and the fluctuations of the possible outliers in the data

characteristic polynomials in real ginibre ensembles we calculate the average of two characteristic polynomials for the real ginibre ensemble of asymmetric random matrices and its chiral counterpart considered as quadratic forms they determine a skew symmetric kernel from which all complex eigenvalue correlations can be derived our results are obtained in a very simple fashion without going to an eigenvalue representation and are completely new in the chiral case they hold for gaussian ensembles which are partly symmetric with kernels given in terms of hermite and laguerre polynomials respectively depending on an asymmetry parameter this allows us to interpolate between the maximally asymmetric real ginibre and the gaussian orthogonal ensemble as well as their chiral counterparts

eigenvalue correlations in non hermitean symplectic random matrices a correlation function of complex eigenvalues of n n random matrices drawn from non hermitean random matrix ensembles of symplectic symmetry is given in terms of a quaternion determinant spectral properties of gaussian ensembles are studied in detail in the regimes of weak and strong non hermiticity

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

induced ginibre ensemble of random matrices and quantum operations a generalization of the ginibre ensemble of non hermitian random square matrices is introduced the corresponding probability measure is induced by the ensemble of rectangular gaussian matrices via a quadratization procedure we derive the joint probability density of eigenvalues for such an induced ginibre ensemble and study various spectral correlation functions for complex and real matrices and analyse universal behaviour in the limit of large dimensions in this limit the eigenvalues of the induced ginibre ensemble cover uniformly a ring in the complex plane the real induced ginibre ensemble is shown to be useful to describe the statistical properties of evolution operators associated with random quantum operations for which the dimensions of the input state and the output state do differ

symplectic structure of the real ginibre ensemble we give a simple derivation of all n point densities for the eigenvalues of the real ginibre ensemble with even dimension n as quaternion determinants a very simple symplectic kernel governs both the real and complex correlations one and two point correlations are discussed in more detail scaling forms for large dimension n are derived

the smallest eigenvalue distribution in the real wishart laguerre ensemble with even topology we consider rectangular random matrices of size p times n belonging to thereal wishart laguerre ensemble also known as the chiral gaussian orthogonalensemble this ensemble appears in many applications like qcd mesoscopicphysics and time series analysis we are particularly interested in thedistribution of the smallest non zero eigenvalue and the gap probability tofind no eigenvalue in an interval 0 t while for odd topology nu n p explicit closed results are known for finite and infinite matrix size for even nu 2 only recursive expressions in p are available the smallest eigenvaluedistribution as well as the gap probability for general even nu isequivalent to expectation values of characteristic polynomials raised to ahalf integer the computation of such averages is done via a combination ofskew orthogonal polynomials and bosonisation methods the results are given interms of pfaffian determinants both at finite p and in the hard edge scalinglimit p to infty and nu fixed for an arbitrary even topology nu numerical simulations for the correlated wishart ensemble illustrate theuniversality of our results in this particular limit these simulations pointto a validity of the hard edge scaling limit beyond the invariant case

a method to calculate correlation functions for β 1 random matrices of odd size the calculation of correlation functions for 1 random matrix ensembles which can be carried out using pfaffians has the peculiar feature of requiring a separate calculation depending on the parity of the matrix size n this same complication is present in the calculation of the correlations for the ginibre orthogonal ensemble of real gaussian matrices in fact the methods used to compute the 1 n odd correlations break down in the case of n odd real ginibre matrices necessitating a new approach to both problems the new approach taken in this work is to deduce the 1 n odd correlations as limiting cases of their n even counterparts when one of the particles is removed towards infinity this method is shown to yield the correlations for n odd real gaussian matrices

truncations of random orthogonal matrices statistical properties of nonsymmetric real random matrices of size m obtained as truncations of random orthogonal n n matrices are investigated we derive an exact formula for the density of eigenvalues which consists of two components finite fraction of eigenvalues are real while the remaining part of the spectrum is located inside the unit disk symmetrically with respect to the real axis in the case of strong nonorthogonality m n const the behavior typical to real ginibre ensemble is found in the case m n l with fixed l a universal distribution of resonance widths is recovered

a new approach to derive pfaffian structures for random matrix ensembles correlation functions for matrix ensembles with orthogonal and unitary symplectic rotation symmetry are more complicated to calculate than in the unitary case the supersymmetry method and the orthogonal polynomials are two techniques to tackle this task recently we presented a new method to average ratios of characteristic polynomials over matrix ensembles invariant under the unitary group here we extend this approach to ensembles with orthogonal and unitary symplectic rotation symmetry we show that pfaffian structures can be derived for a wide class of orthogonal and unitary symplectic rotation invariant ensembles in a unifying way this also includes those for which this structure was not known previously as the real ginibre ensemble and the gaussian real chiral ensemble with two independent matrices as well

skew orthogonal polynomials and the partly symmetric real ginibre ensemble the partly symmetric real ginibre ensemble consists of matrices formed as linear combinations of real symmetric and real anti symmetric gaussian random matrices such matrices typically have both real and complex eigenvalues for a fixed number of real eigenvalues an earlier work has given the explicit form of the joint eigenvalue probability density function we use this to derive a pfaffian formula for the corresponding summed up generalized partition function this pfaffian formula allows the probability that there are exactly k eigenvalues to be written as a determinant with explicit entries it can be used too to give the explicit form of the correlation functions provided certain skew orthogonal polynomials are computed this task is accomplished in terms of hermite polynomials and allows us to proceed to analyze various scaling limits of the correlations including that in which the matrices are only weakly non symmetric