uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory we consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn x dx e nv x dx on the line as n the potentials v are assumed to be real analytic with sufficient growth at infinity the principle results concern plancherel rotach type asymptotics for the orthogonal polynomials down to the axis using these asymptotics we then prove universality for a variety of statistical quantities arising in the theory of random matrix models some of which have been considered recently in 31 and also in 4 an additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials see also 4 the orthogonal polynomial problem is formulated as a riemann hilbert problem following 19 20 the riemann hilbert problem is analyzed in turn using the steepest descent method introduced in 12 and further developed in 11 13 a critical role in our method is played by the equilibrium measure d v for v as analyzed in 8 1999 john wiley sons inc

on the correlation functions of the characteristic polynomials of non hermitian random matrices with independent entries the paper is concerned with the asymptotic behavior of the correlation functions of the characteristic polynomials of non hermitian random matrices with independent entries it is shown that the correlation functions behave like that for the complex ginibre ensemble up to a factor depending only on the fourth absolute moment of the common probability law of the matrix entries

universal behavior for averages of characteristic polynomials at the origin of the spectrum it has been shown by strahov and fyodorov that averages of products and ratios of characteristic polynomials corresponding to hermitian matrices of a unitary ensemble involve kernels related to orthogonal polynomials and their cauchy transforms we will show that for the unitary ensemble open image in new window hermitian matrices these kernels have universal behavior at the origin of the spectrum as n in terms of bessel functions our approach is based on the characterization of orthogonal polynomials together with their cauchy transforms via a matrix riemann hilbert problem due to fokas its and kitaev and on an application of the deift zhou steepest descent method for matrix riemann hilbert problems to obtain the asymptotic behavior of the riemann hilbert problem

universality properties of gelfand tsetlin patterns a standard gelfand tsetlin pattern of depth n is a configuration of particles in 1 ldots n times mathbb r for each r in 1 ldots n r times mathbb r is referred to as the rth level of the pattern a standard gelfand tsetlin pattern has exactly r particles on each level r and particles on adjacent levels satisfy an interlacing constraint probability distributions on the set of gelfand tsetlin patterns of depth n arise naturally as distributions of eigenvalue minor processes of random hermitian matrices of size n we consider such probability spaces when the distribution of the matrix is unitarily invariant prove a determinantal structure for a broad subclass and calculate the correlation kernel in particular we consider the case where the eigenvalues of the random matrix are fixed this corresponds to choosing uniformly from the set of gelfand tsetlin patterns whose nth level is fixed at the eigenvalues of the matrix fixing q n in 1 ldots n and letting n under the assumption that frac q n n to alpha in 0 1 and the empirical distribution of the particles on the nth level converges weakly the asymptotic behaviour of particles on level q n is relevant to free probability theory saddle point analysis is used to identify the set in which these particles behave asymptotically like a determinantal random point field with the sine kernel

average characteristic polynomials in the two matrix model the two matrix model is defined on pairs of hermitian matrices m1 m2 of size n n by the probability measure 1znexp tr v m1 w m2 m1m2 dm1dm2 where v and w are given potential functions and r we study averages of products and ratios of characteristic polynomials in the two matrix model where both matrices m1 and m2 may appear in a combined way in both numerator and denominator we obtain determinantal expressions for such averages the determinants are constructed from several building blocks the biorthogonal polynomials pn x and qn y associated with the two matrix model certain transformed functions pn w and qn v and finally cauchy type transforms of the four eynard mehta kernels k1 1 k1 2 k2 1 and k2 2 in this way we generalize known results for the one matrix model our results also imply a new proof of the eynard mehta theorem for correlation functions in the two matrix model and they lead to a generating function for averages of products of traces

averages of characteristic polynomials in random matrix theory we compute averages of products and ratios of characteristic polynomials associated with orthogonal unitary and symplectic ensembles of random matrix theory the pfaffian determinantal formulae for these averages are obtained and the bulk scaling asymptotic limits are found for ensembles with gaussian weights classical results for the correlation functions of the random matrix ensembles and their bulk scaling limits are deduced from these formulae by a simple computation we employ a discrete approximation method the problem is solved for discrete analogues of random matrix ensembles originating from representation theory and then a limit transition is performed exact pfaffian determinantal formulae for the discrete averages are proven using standard tools of linear algebra no application of orthogonal or skew orthogonal polynomials is needed 2005 wiley periodicals inc

the circular unitary ensemble and the riemann zeta function the microscopic landscape and a new approach to ratios we show in this paper that after proper scalings the characteristic polynomial of a random unitary matrix converges to a random analytic function whose zeros which are on the real line form a determinantal point process with sine kernel our scaling is performed at the so called microscopic level that is we consider the characteristic polynomial at points whose distance to 1 has order 1 n we prove that the rescaled characteristic polynomial does not even have a moment of order one hence making the classical techniques of random matrix theory difficult to apply in order to deal with this issue we couple all the dimensions n on a single probability space in such a way that almost sure convergence occurs when n goes to infinity the strong convergence results in this setup provide us with a new approach to ratios we are able to solve open problems about the limiting distribution of ratios of characteristic polynomials evaluated at points of the form exp 2 i pi alpha n and related objects such as the logarithmic derivative we also explicitly describe the dependence relation for the logarithm of the characteristic polynomial evaluated at several points on the microscopic scale on the number theory side inspired by the work by keating and snaith we conjecture some new limit theorems for the value distribution of the riemann zeta function on the critical line at the level of stochastic processes

scattering reflection and impedance of waves in chaotic and disordered systems with absorption we review recent progress in analysing wave scattering in systems with both intrinsic chaos and or disorder and internal losses when the scattering matrix is no longer unitary by mapping the problem onto a nonlinear supersymmetric model we are able to derive closed form analytic expressions for the distribution of reflection probability in a generic disordered system one of the most important properties resulting from such an analysis is statistical independence between the phase and the modulus of the reflection amplitude in every perfectly open channel the developed theory has far reaching consequences for many quantities of interest including local green functions and time delays in particular we point out the role played by absorption as a sensitive indicator of mechanisms behind the anderson localization transition we also provide a random matrix based analysis of s matrix and impedance correlations for various symmetry classes as well as the distribution of transmitted power for systems with broken time reversal invariance completing previous works on the subject the results can be applied in particular to the experimentally accessible impedance and reflection in a microwave or an ultrasonic cavity attached to a system of antennas

strong asymptotics of orthogonal polynomials with respect to exponential weights we consider asymptotics of orthogonal polynomials with respect to weights w x dx e q x dx on the real line where q x 2m k 0 qkx k q2m 0 denotes a polynomial of even order with positive leading coefficient the orthogonal polynomial problem is formulated as a riemann hilbert problem following 22 23 we employ the steepest descent type method introduced in 18 and further developed in 17 19 in order to obtain uniform plancherel rotach type asymptotics in the entire complex plane as well as asymptotic formulae for the zeros the leading coefficients and the recurrence coefficients of the orthogonal polynomials c 1999 john wiley sons inc

on absolute moments of characteristic polynomials of a certain class of complex random matrices the integer moments of the spectral determinant det zi w 2 of complex random matrices w are obtained in terms of the characteristic polynomial of the positive semidefinite matrix ww for the class of matrices w au where a is a given matrix and u is random unitary this work is motivated by studies of complex eigenvalues of random matrices and potential applications of the obtained results are discussed in this context

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

discrete painleve equations and their appearance in quantum gravity we discuss an algorithmic approach for both deriving discrete analogues of painleve equations as well as using such equations to characterize similarity reductions of spatially discrete integrable evolution equations as a concrete example we show that a discrete analogue of painleve i can be used to characterize similarity solutions of the kac moerbeke equation it turns out that these similarity solutions also satisfy a special case of painleve iv equation in addition we discuss a methodology for obtaining the relevant continuous limits not only at the level of equations but also at the level of solutions as an example we use the wkb method in the presence of two turning points of the third order to parametrize at the continuous limit the solution of painleve i in terms of the solution of discrete painleve i finally we show that these results are useful for investigating the partition function of the matrix model in 2d quantum gravity associated with the measure exp t1z2 t2z4 t3z6

universal shocks in the wishart random matrix ensemble we show that the derivative of the logarithm of the average characteristic polynomial of a diffusing wishart matrix obeys an exact partial differential equation valid for an arbitrary value of n the size of the matrix in the large n limit this equation generalizes the simple inviscid burgers equation that has been obtained earlier for hermitian or unitary matrices the solution through the method of characteristics presents singularities that we relate to the precursors of shock formation in the burgers equation the finite n effects appear as a viscosity term in the burgers equation using a scaling analysis of the complete equation for the characteristic polynomial in the vicinity of the shocks we recover in a simple way the universal bessel oscillations so called hard edge singularities familiar in random matrix theory

an exact formula for general spectral correlation function of random hermitian matrices we have found an exact formula expressing a general correlation function containing both products and ratios of characteristic polynomials of random hermitian matrices the answer is given in the form of a determinant an essential difference from the previously studied correlation functions of products only is the appearance of non polynomial functions along with the orthogonal polynomials these non polynomial functions are the cauchy transforms of the orthogonal polynomials the result is valid for arbitrary ensemble of 2 symmetry class and generalizes recent asymptotic formulae obtained for gaussian unitary ensemble and its chiral counterpart by different methods

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

application of the τ function theory of painleve equations to random matrices piv pii and the gue tracy and widom have evaluated the cumulative distribution of the largest eigenvalue for the finite and scaled infinite gue in terms of a piv and pii transcendent respectively we generalise these results to the evaluation of where for and otherwise and the average is with respect to the joint eigenvalue distribution of the gue as well as to the evaluation of of particular interest are and f n 2 and their scaled limits which give the distribution of the largest eigenvalue and the density respectively our results are obtained by applying the okamoto function theory of piv and pii for which we give a self contained presentation based on the recent work of noumi and yamada we point out that the same approach can be used to study the quantities and f n a for the other classical matrix ensembles

on the second order correlation function of the characteristic polynomial of a hermitian wigner matrix we consider the asymptotics of the second order correlation function of the characteristic polynomial of a random matrix we show that the known result for a random matrix from the gaussian unitary ensemble essentially continues to hold for a general hermitian wigner matrix our proofs rely on an explicit formula for the exponential generating function of the second order correlation function of the characteristic polynomial furthermore we show that the second order correlation function of the characteristic polynomial is closely related to that of the permanental polynomial

on the averages of characteristic polynomials from classical groups we provide an elementary and self contained derivation of formulae for averages of products and ratios of characteristic polynomials of random matrices from classical groups using classical results due to weyl and littlewood

characteristic polynomials of random matrices number theorists have studied extensively the connections between the distribution of zeros of the riemann function and of some generalizations with the statistics of the eigenvalues of large random matrices it is interesting to compare the average moments of these functions in an interval to their counterpart in random matrices which are the expectation values of the characteristic polynomials of the matrix it turns out that these expectation values are quite interesting for instance the moments of order 2k scale for unitary invariant ensembles as the density of eigenvalues raised to the power k 2 the prefactor turns out to be a universal number i e it is independent of the specific probability distribution an equivalent behaviour and prefactor had been found as a conjecture within number theory the moments of the characteristic determinants of random matrices are computed here as limits at coinciding points of multi point correlators of determinants these correlators are in fact universal in dyson s scaling limit in which the difference between the points goes to zero the size of the matrix goes to infinity and their product remains finite

the n level spectral correlations for chaotic systems we study the n level spectral correlation functions of classically chaotic quantum systems without time reversal symmetry according to bohigas giannoni and schmit s universality conjecture it is expected that the correlation functions are in agreement with the prediction of the circular unitary ensemble cue of random matrices a semiclassical resummation formalism allows us to express the correlation functions as sums over pseudo orbits using an extended version of the diagonal approximation on the pseudo orbit sums we derive the n level correlation functions identical to the n n determinantal correlation functions of the cue

level curvature distribution in a model of two uncoupled chaotic subsystems we study distributions of eigenvalue curvatures for a block diagonal random matrix perturbed by a full random matrix the most natural physical realization of this model is a quantum chaotic system with some inherent symmetry such that its energy levels form two independent subsequences subject to a generic perturbation which does not respect the symmetry we describe analytically a crossover in the form of a curvature distribution with a tunable parameter namely the ratio of intersubsystem intrasubsystem coupling strengths we find that the peak value of the curvature distribution is much more sensitive to the changes in this parameter than the power law tail behavior this observation may help to clarify some qualitative features of the curvature distributions observed experimentally in acoustic resonances of quartz blocks

universal results for correlations of characteristic polynomials riemann hilbert approach we prove that general correlation functions of both ratios and products of characteristic polynomials of hermitian random matrices are governed by integrable kernels of three different types a those constructed from orthogonal polynomials b those constructed from cauchy transforms of the same orthogonal polynomials and finally c those constructed from both orthogonal polynomials and their cauchy transforms these kernels are related with the riemann hilbert problem for orthogonal polynomials for the correlation functions we obtain exact expressions in the form of determinants of these kernels derived representations enable us to study asymptotics of correlation functions of characteristic polynomials via the deift zhou steepest descent stationary phase method for riemann hilbert problems and in particular to find negative moments of characteristic polynomials this reveals the universal parts of the correlation functions and moments of characteristic polynomials for an arbitrary invariant ensemble of 2 symmetry class

correlation functions of complex matrix models for a restricted class of potentials harmonic gaussian potentials we express the resolvent integral for the correlation functions of simple traces of powers of complex matrices of size n in terms of a determinant this determinant is a function of four kernels constructed from the orthogonal polynomials corresponding to the potential and from their cauchy transform the correlation functions are a sum of expressions attached to a set of fully packed oriented loop configurations we apply this technique to rotational invariant systems more specifically in the case of the gaussian potential explicit expressions can be written for each configuration and their large n expansion t hooft expansion and their bmn limit can be calculated on simple examples we recover results already known in the literature

robust stability and control for uncertain neutral time delay systems in this article the problem of robust stability and stabilisation for a class of uncertain neutral systems with discrete and distributed time delays is considered by utilising a new lyapunov functional based on the idea of delay partitioning approach we employ the linear matrix inequality technique to derive delay dependent criteria which ensures the robust stability of uncertain neutral systems the obtained stability conditions are formulated in terms of linear matrix inequalities that can easily be solved by using standard software packages further the result is extended to study the robust stabilisation for uncertain neutral systems with parameter uncertainties a state feedback controller is proposed to guarantee the robust asymptotic stabilisation for uncertain systems and the controller is constructed in terms of the solution to a set of matrix inequalities finally numerical examples are presented to illustrate the effectiveness and conservatism of the obtained results it is shown that the

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 correlation functions of the characteristic polynomials of the hermitian sample covariance matrices we consider asymptotic behavior of the correlation functions of the characteristic polynomials of the hermitian sample covariance matrices h n n 1 a m n a m n where am n is a m n complex random matrix with independent and identically distributed entries mathfrak r a alpha j and mathfrak i a alpha j we show that for the correlation function of any even order the asymptotic behavior in the bulk and at the edge of the spectrum coincides with those for the gaussian unitary ensemble up to a factor depending only on the fourth moment of the common probability law of entries mathfrak r a alpha j mathfrak i a alpha j i e the higher moments do not contribute to the above limit

mixed correlation function and spectral curve for the 2 matrix model we compute the mixed correlation function in a way which involves only the orthogonal polynomials with degrees close to n in some sense like the christoffel darboux theorem for non mixed correlation functions we also derive new representations for the differential systems satisfied by the bi orthogonal polynomials and we find new formulae for the spectral curve in particular we prove the conjecture of m bertola claiming that the spectral curve is the same curve which appears in the loop equations

on statistics of bi orthogonal eigenvectors in real and complex ginibre ensembles combining partial schur decomposition with supersymmetry we suggest a method of studying the joint probability density jpd of an eigenvalue and the associated non orthogonality overlap factor also known as the eigenvalue condition number of the left and right eigenvectors for non selfadjoint gaussian random matrices of size n times n first we derive the general finite n expression for the jpd of a real eigenvalue lambda and the associated non orthogonality factor in the real ginibre ensemble and then analyze its bulk and edge scaling limits the ensuing distribution is maximally heavy tailed so that all integer moments beyond normalization are divergent a similar calculation for a complex eigenvalue z and the associated non orthogonality factor in the complex ginibre ensemble is presented as well and yields a distribution with the finite first moment its bulk scaling limit yields a distribution whose first moment reproduces the well known result of chalker and mehlig phys rev lett 81 16 3367 3370 1998 and we provide the edge scaling distribution for this case as well our method involves evaluating the ensemble average of products and ratios of integer and half integer powers of characteristic polynomials for ginibre matrices which we perform in the framework of a supersymmetry approach our paper complements recent studies by bourgade and dubach the distribution of overlaps between eigenvectors of ginibre matrices 2018 arxiv 1801 01219

universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles this paper is devoted to the rigorous proof of the universality conjecture of random matrix theory according to which the limiting eigenvalue statistics ofn n random matrices within spectral intervals ofo n 1 is determined by the type of matrix real symmetric hermitian or quaternion real and by the density of states we prove this conjecture for a certain class of the hermitian matrix ensembles that arise in the quantum field theory and have the unitary invariant distribution defined by a certain function the potential in the quantum field theory satisfying some regularity conditions

quantum limit of the laser line width in chaotic cavities and statistics of residues of scattering matrix poles he quantum limited line width of a laser cavity is enhanced above the schawlow townes value by the petermann factor k due to the non orthogonality of the cavity modes we derive the relation between the petermann factor and the residues of poles of the scattering matrix and investigate the statistical properties of the petermann factor for cavities in which the radiation is scattered chaotically for a single scattering channel we determine the complete probability distribution of k and find that the average petermann factor depends non analytically on the area of the opening and greatly exceeds the most probable value for an arbitrary number n of scattering channels we calculate as a function of the decay rate i of the lasing mode we find for n 1 that for typical values of i the average petermann factor propto sqrt n 1 is parametrically larger than unity

moments of the characteristic polynomial in the three ensembles of random matrices moments of the characteristic polynomial of a random matrix taken from any of the three ensembles orthogonal unitary or symplectic are given either as a determinant or a pfaffian or as a sum of determinants for gaussian ensembles on comparing two expressions of the same moment one obtains two remarkable identities one between an n n determinant and an m m determinant and another between the pfaffian of a 2n 2n anti symmetric matrix and a sum of m m determinants

random matrices and l functions in recent years there has been a growing interest in connections between the statistical properties of number theoretical l functions and random matrix theory we review the history of these connections some of the major achievements and a number of applications

semiclassical asymptotics of orthogonal polynomials riemann hilbert problem and universality in the matrix model we derive semiclassical asymptotics for the orthogonal polynomials pn z on the line with respect to the exponential weight exp inv z where v z is a double well quartic polynomial in the limit when n n 1 we assume that n n cri for some 0 where cr is the critical value which separates orthogonal polynomials with two cuts from the ones with one cut simultaneously we derive semiclassical asymptotics for the recursive coe cients of the orthogonal polynomials and we show that these coe cients form a cycle of period two which drifts slowly with the change of the ratio n n the proof of the semiclassical asymptotics is based on the methods of the theory of integrable systems and on the analysis of the appropriate matrix riemannhilbert problem as an application of the semiclassical asymptotics of the orthogonal polynomials we prove the universality of the local distribution of eigenvalues in the matrix model with the double well quartic interaction in the presence of two cuts contents

principal component analysis of persistent homology rank functions with case studies of spatial point patterns sphere packing and colloids abstract persistent homology while ostensibly measuring changes in topology captures multiscale geometrical information it is a natural tool for the analysis of point patterns in this paper we explore the statistical power of the persistent homology rank functions for a point pattern x we construct a filtration of spaces by taking the union of balls of radius a centred on points in x x a x x b x a the rank function k x a b r 2 a b r is then defined by k x a b rank h k x a h k x b where is the induced map on homology from the inclusion map on spaces we consider the rank functions as lying in a hilbert space and show that under reasonable conditions the rank functions from multiple simulations or experiments will lie in an affine subspace this enables us to perform functional principal component analysis which we apply to experimental data from colloids at different effective temperatures and to sphere packings with different volume fractions we also investigate the potential of rank functions in providing a test of complete spatial randomness of 2d point patterns using the distances to an empirically computed mean rank function of binomial point patterns in the unit square

the law of large numbers for the maximum of almost gaussian log correlated fields coming from random matrices we compute the leading asymptotics as n rightarrow infty of the maximum of the field q n q log q a n q in mathbb c for any unitarily invariant hermitian random matrix a n associated to a non critical real analytic potential hence we verify the leading order in a conjecture of fyodorov and simm nonlinearity 29 2837 2016 arxiv 1503 07110 math ph formulated for the gue the method relies on a classical upper bound and a more sophisticated lower bound based on a variant of the second moment method which exploits the hyperbolic branching structure of the field q n q q in mathbb h specifically we compare q n to an idealized gaussian field by means of exponential moments in principle this method could also be applied to random fields coming from other point processes provided that one can compute certain mixed exponential moments for unitarily invariant ensembles we show that these assumptions follow from the fyodorov strahov formula fyodorov and strahov in j phys a 36 12 3203 3213 2003 https doi org 10 1088 0305 4470 36 12 320 and asymptotics of orthogonal polynomials derived in deift et al commun pure appl math 52 11 1335 1425 1999 https doi org 10 1002 sici 1097 0312 199911 52 11 3 0 co 2 1

random matrix theory and ζ 1 2 it we study the characteristic polynomials z u of matrices u in the circular unitary ensemble cue of random matrix theory exact expressions for any matrix size n are derived for the moments of z and z z and from these we obtain the asymptotics of the value distributions and cumulants of the real and imaginary parts of log z as n in the limit we show that these two distributions are independent and gaussian costin and lebowitz 15 previously found the gaussian limit distribution for im log z using a different approach and our result for the cumulants proves a conjecture made by them in this case we also calculate the leading order n asymptotics of the moments of z and z z these cue results are then compared with what is known about the riemann zeta function s on its critical line re s 1 2 assuming the riemann hypothesis equating the mean density of the non trivial zeros of the zeta function at a height t up the critical line with the mean density of the matrix eigenvalues gives a connection between n and t invoking this connection our cue results coincide with a theorem of selberg for the value distribution of log 1 2 it in the limit t they are also in close agreement with numerical data computed by odlyzko 29 for large but finite t this leads us to a conjecture for the moments of 1 2 it finally we generalize our random matrix results to the circular orthogonal coe and circular symplectic cse ensembles