critical statistics for non hermitian matrices we introduce a generalized ensemble of non hermitian matrices interpolating between the gaussian unitary ensemble the ginibre ensemble and the poisson ensemble the joint eigenvalue distribution of this model is obtained by means of an extension of the itzykson zuber formula to general complex matrices its correlation functions are studied both in the case of weak non hermiticity and in the case of strong non hermiticity in the weak non hermiticity limit we show that the spectral correlations in the bulk of the spectrum display critical statistics the asymptotic linear behavior of the number variance is already approached for energy differences of the order of the eigenvalue spacing to lowest order its slope does not depend on the degree of non hermiticity close the edge the spectral correlations are similar to the hermitian case in the strong non hermiticity limit the crossover behavior from the ginibre ensemble to the poisson ensemble first appears close to the surface of the spectrum our model may be relevant for the description of the spectral correlations of an open disordered system close to an anderson transition

gap probabilities in non hermitian random matrix theory we compute the gap probability that a circle of radius r around the origin contains exactly k complex eigenvalues four different ensembles of random matrices are considered the ginibre ensembles and their chiral complex counterparts with both complex 2 or quaternion real 4 matrix elements for general non gaussian weights we give a fredholm determinant or pfaffian representation respectively depending on the non hermiticity parameter at maximal non hermiticity that is for rotationally invariant weights the product of fredholm eigenvalues for 4 follows from the 2 case by skipping every second factor in contrast to the known relation for hermitian ensembles on additionally choosing gaussian weights we give new explicit expressions for the fredholm eigenvalues in the chiral case in terms of bessel k and incomplete bessel i functions this compares with known results for the ginibre ensembles in terms of incomplete exponentials furthermore we present an asymptotic expansion of the log

on the bezoutian for polynomial matrices abstract the bezoutian b of two polynomial matrices can be described as a solution of a linear matrix equation this fact yields a new proof of the barnett factorization of b

truncations of random unitary matrices we analyse properties of non hermitian matrices of size m constructed as square submatrices of unitary orthogonal random matrices of size n m distributed according to the haar measure in this way we define ensembles of random matrices and study the statistical properties of the spectrum located inside the unit circle in the limit of large matrices this ensemble is characterized by the ratio m n for the truncated cue we analytically derive the joint density of eigenvalues and all correlation functions in the strongly non unitary case universal ginibre behaviour is found for n m fixed and n to the universal resonance width distribution with n m open channels is recovered

transition between hermitian and non hermitian gaussian ensembles the transition between hermitian and non hermitian matrices of the gaussian unitary ensemble is revisited an expression for the kernel of the rescaled hermite polynomials is derived which expresses the sum in terms of the highest order polynomials from this christoffel darboux like formula some results are derived including an extension to the complex plane of the airy kernel

s matrix poles for chaotic quantum systems as eigenvalues of complex symmetric random matrices from isolated to overlapping resonances we study complex eigenvalues of large symmetric random matrices of the form where both and are real symmetric is a random gaussian and is such that when when the model can be used to describe the universal statistics of s matrix poles resonances in the complex energy plane we derive the ensuing distribution of the resonance widths which generalizes the well known distribution to the case of overlapping resonances we also consider a different class of almost real matrices when is random and uncorrelated with

statistical properties of eigenvectors in non hermitian gaussian random matrix ensembles statistical properties of eigenvectors in non hermitian random matrix ensembles are discussed with an emphasis on correlations between left and right eigenvectors two approaches are described one is an exact calculation for ginibre s ensemble in which each matrix element is an independent identically distributed gaussian complex random variable the other is a simpler calculation using n 1 as an expansion parameter where n is the rank of the random matrix this is applied to girko s ensemble consequences of eigenvector correlations which may be of physical importance in applications are also discussed it is shown that eigenvalues are much more sensitive to perturbations than in the corresponding hermitian random matrix ensembles it is also shown that in problems with time evolution governed by a non hermitian random matrix transients are controlled by eigenvector correlations

fluctuation formula for complex random matrices a gaussian fluctuation formula is proved for linear statistics of complex random matrices in the case where the statistic is rotationally invariant for a general linear statistic without this symmetry coulomb gas theory is used to predict that the distribution will again be a gaussian with a specific mean and variance the variance splits naturally into a bulk and surface contibution the latter resulting from the long range correlations at the boundary of the support of the eigenvalue density

eigenfunction statistics of complex systems a common mathematical formulation we derive a common mathematical formulation for the eigenfunction statistics of hermitian operators represented by a multiparametric probability density the system information in the formulation enters through two parameters only namely system size and the complexity parameter a function of all system parameters including size the behavior is contrary to the eigenvalue statistics which is sensitive to the complexity parameter only and shows a single parametric scaling the existence of a mathematical formulation of both eigenfunctions and eigenvalues common to a wide range of complex systems indicates the possibility of a similar formulation for many physical properties this also suggests the possibility to classify them in various universality classes defined by the complexity parameter

bezoutians and intertwining matrices abstract using the notion of an intertwining matrix introduced in a previous paper the authors show that the class of all n by n bezoutians compatible with a nonzero polynomial of degree n may be characterized by an intertwining relation as well as by four other conditions formulated in terms of the structure of these matrices

eigenvalue spacing statistics of a four matrix model of some four by four random matrices a simple four matrix model consisting of some 4 4 real and imaginary random matrices and three parameters is introduced it is shown that the eigenvalue spacing statistics of an ensemble of such matrices can be used to describe transitions between all the wigner surmises of random matrix theory formulae for the nearest neighbour spacing distributions of the ensemble are given for various parameter values

the solution of a chiral random matrix model with complex eigenvalues we describe in detail the solution of the extension of the chiral gaussian unitary ensemble chgue into the complex plane the correlation functions of the model are first calculated for a finite number of n complex eigenvalues where we exploit the existence of orthogonal laguerre polynomials in the complex plane when taking the large n limit we derive new correlation functions in the case of weak and strong non hermiticity thus describing the transition from the chgue to a generalized ginibre ensemble we briefly discuss applications to the dirac operator eigenvalue spectrum in quantum chromodynamics with non vanishing chemical potential this is an extended version of hep th 0204068

random matrices beyond the cartan classification it is known that hermitean random matrix ensembles can be identified with symmetric coset spaces of lie groups or else with tangent spaces of the same this results in a classification of random matrix ensembles as well as applications in practical calculations of physical observables in this paper we show that a large number of non hermitean random matrix ensembles defined by physically motivated symmetries chiral symmetry time reversal invariance space rotation invariance particle hole symmetry or different reality conditions can likewise be identified with symmetric spaces we give explicit representations of the random matrix ensembles identified with lateral algebra subspaces and of the corresponding symmetric subalgebras spanning the group of invariance among the ensembles listed we identify as special cases all the hermitean ensembles identified with the cartan classes of symmetric spaces and the three ginibre ensembles with complex eigenvalues

the kernel of the bezoutian of polynomial matrices abstract a definition of anderson and jury is extended to define a bezoutian b for a quadruple f g u d of polynomial matrices which have the property that the strictly proper rational parts of f 1 g and ud 1 are equal the right kernel of b is described by means of block hankel matrices associated to d and to a greatest common right divisor of d and u

non hermitian random matrix theory summation of planar diagrams the single ring theorem and the disc annulus phase transition i review aspects of work done in collaboration with a zee and r scalettar 1997 nucl phys b 504 579 1997 nucl phys b 501 643 2001 j math phys 42 5718 on complex non hermitian random matrices i open by explaining why the bag of tools used regularly in analysing hermitian random matrices cannot be applied directly to analyse non hermitian matrices and then introduce the method of hermitization which solves this problem then for rotationally invariant ensembles i derive a master equation for the average density of eigenvalues in the complex plane in the limit of infinitely large matrices this is achieved by resumming all the planar diagrams which appear in the perturbative expansion of the hermitized green function remarkably this resummation can be carried out explicitly for any rotationally invariant ensemble i prove that in the limit of infinitely large matrices the shape of the eigenvalue distribution is either a disc or an annulus this is the celebrated single ring theorem which of these shapes is realized is determined by the parameters coupling constants which determine the ensemble by varying these parameters a phase transition may occur between the two possible shapes i briefly discuss the universal features of this transition as the analysis of this problem relies heavily on summation of planar feynman diagrams i make a special effort at presenting a pedagogical exposition of the diagrammatic method which some readers may find useful

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

how to fit the degree distribution of the air network abstract we investigate three different approaches for fitting the degree distributions of china us and the composite china us air network in order to reveal the nature of such distributions and the potential theoretical background on which they are based our first approach is the fitting with q statistics probability distribution done separately in two regimes this yields acceptable outcomes but generates two sets of fitting parameters the second approach is an entire fitting to all data points with the formula proposed by tsallis et al so far this trial is not able to produce consistent results in the third approach we fit the data with two composite distributions which may lack theoretical support for the moment

conformal deformation of equilibrium measures in normal random ensembles we investigate the eigenvalue statistics of ensembles of normal random matrices when their order n tends to infinite in the model the eigenvalues have uniform density within a region determined by a simple analytic polynomial curve we study the conformal deformations of equilibrium measures of normal random ensembles to the real line and give sufficient conditions for it to weakly converge to a wigner measure

topology of technology graphs small world patterns in electronic circuits recent theoretical studies and extensive data analyses have revealed a common feature displayed by biological social and technological networks the presence of small world patterns here we analyze this problem by using several graphs obtained from one of the most common technological systems electronic circuits it is shown that both analogic and digital circuits exhibit small world behavior we conjecture that the small world pattern arises from the compact design in which many elements share a small close physical neighborhood plus the fact that the system must define a single connected component which requires shortcuts connecting different integrated clusters the degree distributions displayed are consistent with a conjecture concerning the sharp cutoffs associated to the presence of costly connections amaral et al proc natl acad sci usa 97 11 149 2000 thus providing a limit case for the classes of universality of small world patterns from real artificial networks the consequences for circuit design are outlined

approach to general methods for fitting and their sensitivity abstract this paper examines the performance of various fitting methods between the curves of two explicit functions the approximating test function is considered to be one parametrical first and multi parametrical directly after the widely used least square deviations constitutional method based on the euclidean norm is not unique methods based on q norm for q 1 can also be defined emphasis on these methods especially for q 1 is placed furthermore any functional fulfilling the norm s preconditions induces a metric for deviations supporting a respective method for fitting through the minimization of total deviations this dissertation addresses also the sensitivity of each method that is a measure of how abrupt the variation of the total deviations near its minimum is we show that the least square method does not indispensably perform the largest sensitivity in regard to the alternative methods based on other q norms in addition we represent the explicit general expression of normal equations from which the fitting can be achieved and the sensitivity from which one can positively extract the suitable norm for a given model

level dynamics for conservative and dissipative quantum systems we establish level dynamics for finite matrices employing a unified treatment of real symmetric complex hermitian quaternion real unitary and arbitrary complex matrices in all cases the level dynamics take the form of the classical hamiltonian flow of some fictitious many particle systems equilibrium statistical mechanics of the latter leads to the well known matrix ensembles of random matrix theory ginibre s ensemble in particular is thus associated with level dynamics of arbitrary complex matrices

integral formulae for the eigenvalue density of complex random matrices we show that the density of complex eigenvalues for unitary invariant ensembles of complex matrices a can be written as an integral over the eigenvalues gj of aa for the standard random matrix ensembles with matrix density of the form jw gj this integral can be further reduced to a twofold integral involving the christoffel darboux kernel for the orthogonal polynomials associated with weight w

bezoutians of polynomial matrices and their generalized inverses abstract the bezoutian b of a quadruple f g u d of polynomial matrices is studied it is shown that under certain coprimeness conditions there is a 1 or a 1 2 inverse of b which is a block hankel matrix

bezoutians of rational matrix functions matrix equations and factorizations abstract the notion of bezoutian for a quadruple of rational matrix functions introduced by the authors in previous works is shown to be an adequate connecting link between certain factorization problems for rational matrix functions and quadratic and linear matrix equations particular attention is given to various types of coprime factorizations a characterization of bezoutians as angle operators between certain pairs of subspaces is also given

intertwining and testing matrices corresponding to a polynomial abstract given a natural number n and a polynomial of degree m n a description is given of a class r n of n by n matrices the intertwining matrices for with the following properties let h n be the family of all n by n hankel matrices compatible with and consider the relation hr t rh then an n by n matrix h belongs to h n if and only if this relation holds for all r r n a characterization is also given of testing matrices a matrix r is said to be a testing matrix if the fulfillment of the above relation already implies that h h n

random matrices close to hermitian or unitary overview of methods and results the paper discusses recent progress in understanding statistical properties of eigenvalues of weakly non hermitian and non unitary random matrices the first type of ensembles is of the form i with being a large random n n hermitian matrix with independent entries deformed by a certain anti hermitian n n matrix i satisfying in the limit of large dimension n the condition tr 2 n tr 2 here can be either a random or just a fixed given hermitian matrix ensembles of such a type with 0 emerge naturally when describing quantum scattering in systems with chaotic dynamics and serve to describe resonance statistics related models are used to mimic complex spectra of the dirac operator with chemical potential in the context of quantum chromodynamics ensembles of the second type arising naturally in scattering theory of discrete time systems are formed by n n matrices with complex entries such that i for 0 this coincides with the circular unitary ensemble and 0 i describes deviation from unitarity our result amounts to answering statistically the following old question given the singular values of a matrix describe the locus of its eigenvalues we systematically show that the obtained expressions for the correlation functions of complex eigenvalues describe a non trivial crossover from wigner dyson statistics of real unimodular eigenvalues typical of hermitian unitary matrices to ginibre statistics in the complex plane typical of ensembles with strong non hermiticity tr 2 tr 2 when n finally we discuss scarce results available on eigenvector statistics for weakly non hermitian random matrices

extending the notions of companion and infinite companion to matrix polynomials the notion of infinite companion matrix is extended to the case of matrix polynomials including polynomials with singular leading coefficient for row reduced polynomials a finite companion is introduced as the compression of the shift matrix the methods are based on ideas of dilation theory connections with systems theory are indicated applications to the problem of linearization of matrix polynomials solution of systems of difference and differential equations and new factorization formulae for infinite block hankel matrices having finite rank are shown as a consequence any system of linear difference or differential equations with constant coefficients can be transformed into a first order system of dimension n deg det d

almost hermitian random matrices crossover from wigner dyson to ginibre eigenvalue statistics by using the method of orthogonal polynomials we analyze the statistical properties of complex eigenvalues of random matrices describing a crossover from hermitian matrices characterized by the wigner dyson statistics of real eigenvalues to strongly non hermitian ones whose complex eigenvalues were studied by ginibre two point statistical measures as e g spectral form factor number variance and small distance behavior of the nearest neighbor distance distribution p s are studied in more detail in particular we found that the latter function may exhibit unusual behavior p s ensuremath propto s 5 2 for some parameter values

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

on the history of the bezoutian and the resultant matrix abstract it is shown how the bezoutian and the resultant matrix evolved from euler s work in elimination theory

large scale correlations in normal non hermitian matrix ensembles we compute the large scale macroscopic correlations in ensembles of normal random matrices with a general non gaussian measure and in ensembles of general non hermitian matrices with a class of non gaussian measures in both cases the eigenvalues are complex and in the large n limit they occupy a domain in the complex plane for the case when the support of eigenvalues is a connected compact domain we compute two three and four point connected correlation functions in the first non vanishing order in 1 n in a manner that the algorithm of computing higher correlations becomes clear the correlation functions are expressed through the solution of the dirichlet boundary problem in the domain complementary to the support of eigenvalues

bezout and hankel matrices associated with row reduced matrix polynomials barnett type formulas abstract based on the approach introduced by b d o anderson and e i jury in 1976 the definition of finite hankel and bezout matrices corresponding to matrix polynomials is extended to the case where the denominator of the corresponding rational matrix function is not necessarily monic but is row reduced the matrices introduced keep most of the well known properties that hold in the monic case in particular we derive extensions of formulas giving a connection with polynomials in the companion matrix usually called barnett formulas of the inversion theorem and of formulas concerning alternating products of hankel and bezout matrices

generalized bezoutian and matrix equations abstract a natural generalization of the classical bezout matrix of two polynomials is introduced for a family of several matrix polynomials the main aim of the paper is to show that this generalized bezoutian serves as an adequate connecting link between the class of equations in matrix polynomials m y z l r and the class of linear matrix equations ax xb c each equation in one of these classes is coupled with a certain equation in the other class so that for each couple the generalized bezoutian corresponding to a solution y z of the equation in matrix polynomials is a solution of the matrix equation and conversely any solution x of the matrix equation is a generalized bezoutian corresponding to a certain solution of the equation in matrix polynomials in particular either both equations are solvable or both have no solutions explicit formulas connecting the solutions of the two equations are given also various representation formulas for the generalized bezoutian are derived and its relation to the resultant matrix and the greatest common divisor of several matrix polynomials is discussed

bezoutian and schur cohn problem for operator polynomials abstract a generalized bezout operator bezoutian for a pair of operator polynomials is introduced and its kernel is described in terms of common spectral data of the underlying polynomials the location of the spectrum of an operator polynomial with compact spectrum with respect to the unit circle infinite dimensional version of the schur cohn problem is expressed via the inertia of a suitable bezoutian an application to the geometric dichotomy problem for difference equations with operator coefficients is given as well

integrable structure of ginibre s ensemble of real random matrices and a pfaffian integration theorem in the recent publication e kanzieper and g akemann in phys rev lett 95 230201 2005 an exact solution was reported for the probability pn k to find exactly k real eigenvalues in the spectrum of an n n real asymmetric matrix drawn at random from ginibre s orthogonal ensemble ginoe in the present paper we offer a detailed derivation of the above result by concentrating on the proof of the pfaffian integration theorem the key ingredient of our analysis of the statistics of real eigenvalues in the ginoe we also initiate a study of the correlations of complex eigenvalues and derive a formula for the joint probability density function of all complex eigenvalues of a ginoe matrix restricted to have exactly k real eigenvalues in the particular case of k 0 all correlation functions of complex eigenvalues are determined

on bezoutians of nonsquare matrix polynomials and inversion of matrices with nonsquare blocks abstract the notion of bezoutian of nonsquare matrix polynomials is defined it is used to establish formulae for the inverse of square matrices partitioned into nonsquare blocks in particular a generalization of the gohberg semencul formula for the inverse of a generalized toeplitz matrix is proved

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

interpolation between airy and poisson statistics for unitary chiral non hermitian random matrix ensembles we consider a family of chiral non hermitian gaussian random matrices in the unitarily invariant symmetry class the eigenvalue distribution in this model is expressed in terms of laguerre polynomials in the complex plane these are orthogonal with respect to a non gaussian weight including a modified bessel function of the second kind and we give an elementary proof for this in the large n limit the eigenvalue statistics at the spectral edge close to the real axis are described by the same family of kernels interpolating between airy and poisson that was recently found by one of the authors for the elliptic ginibre ensemble we conclude that this scaling limit is universal appearing for two different non hermitian random matrix ensembles with unitary symmetry as a second result we give an equivalent form for the interpolating airy kernel in terms of a single real integral similar to representations for the asymptotic kernel in the bulk and at the hard edge of the spectrum this makes its structure as a one parameter deformation of the airy kernel more transparent