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

on the quasilinear elliptic problems with critical sobolev hardy exponents and hardy terms abstract let r n be a smooth bounded domain such that 0 n 3 in this paper we study the critical quasilinear elliptic problems p u u p 2 u x p u p t 2 x t u u q 2 x s u u w 0 1 p with dirichlet boundary condition where p u div u p 2 u 1 p n 0 n p p p 0 0 s t p p q p s p n s n p p t p n t n p p s and p t are the critical sobolev hardy exponents via variational methods we deal with the conditions that ensure the existence of positive solutions for the equation the results depend crucially on the parameters p q s and

the structural property of a class of vector valued hyperbolic equations and applications abstract in this paper we study the structural properties of a class of vector valued hyperbolic equations with appropriate boundary conditions including the spectrum determined growth condition we prove that the equations associate with a c 0 semigroup by the structural analysis we obtained a sufficient and necessary condition for being at least an eigenvalue on the imaginary axis in particular using the asymptotic analysis technique we prove that the spectrum of the operator determined by the equations is distributed in a strip parallel to the imaginary axis and is union of finitely many separable sets furthermore we prove that the root vectors of the operator are complete and there is a sequence of root vectors that forms a riesz basis with parentheses for the hilbert state space as applications of our results we give some concrete examples in controlled complex network of euler bernoulli beams

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

symmetric solutions for a class of singular biharmonic elliptic systems involving critical exponents this paper deals with a class of singular biharmonic elliptic systems involving critical exponents in a bounded symmetric domain by using the variational method and the symmetric criticality principle of palais we obtain several existence and multiplicity results of g symmetric solutions for the systems

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

solutions to critical elliptic equations with multi singular inverse square potentials abstract let be an open bounded domain in r n n 3 with smooth boundary we are concerned with the multi singular critical elliptic problem u i 1 k i u x a i 2 q x u 2 2 u u h 0 1 where i r 2 2 n n 2 a i 1 i k and q x is a positive bounded function on using moser iteration we prove the asymptotic behavior of solutions for at points a i 1 i k by exploiting the effect of the coefficient of the critical nonlinearity we by means of a variational method establish the existence of positive and sign changing solutions for problem

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

existence of solutions for singular elliptic systems with critical exponents abstract let 0 be an open bounded domain in r n n 3 and 2 2 n n 2 we consider the following elliptic system of two equations in h 0 1 h 0 1 u t u x 2 2 u 2 u v u v t v x 2 2 u v 2 v v where 0 and 1 satisfy 2 using moser iteration we prove the asymptotic behavior of solutions for at the origin by exploiting the mountain pass theorem we establish the existence of solutions to we also discuss sign changing solutions for in this paper

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

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

elliptic systems involving the critical exponents and potentials abstract in this paper we investigate a singular elliptic system which involves the critical sobolev exponent and multiple hardy type terms by employing variational methods the existence of its positive solutions is established by the moser iteration method some asymptotic properties of its nontrivial solutions at the singular points are verified

elliptic systems involving critical nonlinearities and different hardy type terms abstract in this paper a singular system of elliptic equations is investigated which involves critical nonlinearities and deferent hardy type terms existence of positive solutions to the system is verified by variational methods and asymptotic properties of solutions at the singular point are established by the moser iteration method it is found that the functions in solutions have different singularities at the singular point

a note on the sign changing solutions to elliptic problems with critical sobolev and hardy terms let rn be a bounded domain such that 0 n 7 2 2nn 2 we obtain existence of sign changing solutions for the dirichlet problem u u x 2 u 2 2u u on u 0 on for suitable positive numbers and

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

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

on the singular elliptic systems involving multiple critical sobolev exponents in this paper a singular elliptic system is investigated which involves multiple critical sobolev exponents and hardy type terms by using variational methods and analytical techniques the existence of positive and sign changing solutions to the system is established

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

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

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

quasilinear elliptic problems with critical exponents and hardy terms abstract let 0 be an open bounded domain r n n p 2 we are concerned with the multiplicity of positive solutions of p u u p 2 u x p u p 2 u q x u p 2 u u w 0 1 p where p u div u p 2 u 1 p n p np n p 0 n p p p 0 and q x is a nonnegative function on by investigating the effect of the coefficient of the critical nonlinearity we by means of variational method prove the existence of multiple positive solutions

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