a few remarks on colour flavour transformations truncations of random unitary matrices berezin reproducing kernels and selberg type integrals we investigate diverse relations of the colour flavour transformations cft introduced by zirnbauer in supersymmetry for systems with unitary disorder circular ensembles 1996 j phys a math gen 29 7113 36 to various topics in random matrix theory and multivariate analysis such as measures on truncations of unitary random matrices jacobi ensembles of random matrices berezin reproducing kernels and a generalization of the selberg integral due to kaneko kadell and yan involving the schur functions apart from suggesting explicit formulae for bosonic cft for the unitary group in the range of parameters beyond that in zirnbauer s paper we also suggest an alternative variant of the transformation with integration going over an unbounded domain of a pair of hermitian matrices the latter makes possible the evaluation of certain averages in random matrix theory

how many eigenvalues of a random matrix are real let a be an n x n matrix whose elements are independent random variables with standard normal distributions as n oo the expected number of real eigenvalues is asymptotic to v 7r we obtain a closed form expres sion for the expected number of real eigenvalues for finite n and a formula for the density of a real eigenvalue for finite n asymptotically a real normalized eigenvalue al ii of such a random matrix is uniformly distributed on the in terval 1 1 analogous but strikingly different results are presented for the real generalized eigenvalues we report on numerical experiments confirming these results and suggesting that the assumption of normality is not important for the asymptotic results department of mathematics massachusetts institute of technology cambridge massachusetts 02139 e mail address edelmanfmath mit edu arts and sciences kapi olani community college 4303 diamond head road honolulu hawaii 98616 e mail address kostlanfkahuna math hawaii edu tj watson rsearch center 32 2 ibm yorktown heights new york 10598 0218 e mail address shubqwatson ibm com this content downloaded from 157 55 39 29 on tue 12 apr 2016 08 54 01 utc all use subject to http about jstor org terms

moments of ratios of characteristic polynomials of a certain class of random matrices we derive a new method of calculating the mean negative moments and ratios of squared modulus of characteristic polynomials of a certain class of random matrices new results obtained with this method are presented

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

derivation of an eigenvalue probability density function relating to the poincare disk a result of zyczkowski and sommers 2000 j phys a math gen 33 2045 57 gives the eigenvalue probability density function for the top n x n sub block of a haar distributed matrix from u n n in the case n n we rederive this result starting from knowledge of the distribution of the sub blocks introducing the schur decomposition and integrating over all variables except the eigenvalues the integration is done by identifying a recursive structure which reduces the dimension this approach is inspired by an analogous approach which has been recently applied to determine the eigenvalue probability density function for random matrices a 1 b where a and b are random matrices with entries standard complex normals we relate the eigenvalue distribution of the sub blocks to a many body quantum state and to the one component plasma on the pseudosphere

route to chaos and some properties in the boundary crisis of a generalized logistic mapping abstract a generalization of the logistic map is considered showing two control parameters and that can reproduce different logistic mappings including the traditional second degree logistic map cubic quartic and all other degrees we introduce a parametric perturbation such that the original logistic map control parameter r changes its value periodically according an additional parameter 2 q the value of q gives this period for this system an analytical expression is obtained for the first bifurcation that starts a period doubling cascade and using the feigenbaum universality we found numerically the accumulation point r c where the cascade finishes giving place to chaos in the second part of the paper we study the death of this chaotic behavior due to a boundary crisis at the boundary crisis orbits can reach a maximum value x x max 1 when it occurs the trajectory is mapped to a fixed point at x 0 we show that there exist a general recursive formula for initial conditions that lead to x x max

random matrix theory and l functions at s 1 2 recent results of katz and sarnak 8 9 suggest that the low lying zeros of families of l functions display the statistics of the eigenvalues of one of the compact groups of matrices u n o n or usp 2n we here explore the link between the value distributions of the l functions within these families at the central point s 1 2 and those of the characteristic polynomials z u of matrices u with respect to averages over so 2n and usp 2n at the corresponding point 0 using techniques previously developed for u n in 10 for any matrix size n we find exact expressions for the moments of z u 0 for each ensemble and hence calculate the asymptotic large n value distributions for z u 0 and log z u 0 the asymptotic results for the integer moments agree precisely with the few corresponding values known for l functions the value distributions suggest consequences for the non vanishing of l functions at the central point

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

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

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

supersymmetry for systems with unitary disorder circular ensembles a generalized hubbard stratonovitch transformation relating an integral over random unitary matrices to an integral over efetov s unitary model manifold is introduced this transformation adapts the supersymmetry method to disordered and chaotic systems that are modelled not by a hamiltonian but by their scattering matrix or time evolution operator in contrast to the standard method no saddle point approximation is made and no massive modes have to be eliminated this first paper on the subject applies the generalized hubbard stratonovitch transformation to dyson s circular unitary ensemble it is shown how a supersymmetric variant of the harish chandra itzykson zuber formula can be used to compute in the large n limit the n level correlation function for any n non trivial applications to random network models quantum chaotic maps and lattice gauge theory are expected

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

on the generation of exact solutions for evaluating numerical schemes and estimating discretization error abstract in this paper we further develop the method of nearby problems mnp for generating exact solutions to realistic partial differential equations by extending it to two dimensions we provide an extensive discussion of the 2d spline fitting approach which provides ck continuity continuity of the solution value and its first k derivatives along spline boundaries and is readily extendable to higher dimensions a detailed one dimensional example is given to outline the general concepts then the two dimensional spline fitting approach is applied to two problems heat conduction with a distributed source term and the viscous incompressible flow in a lid driven cavity with both a constant lid velocity and a regularized lid velocity which removes the strong corner singularities the spline fitting approach results in very small spline fitting errors for the heat conduction problem and the regularized driven cavity whereas the fitting errors in the standard lid driven cavity case are somewhat larger due to the singular behaviour of the pressure near the driven lid the mnp approach is used successfully as a discretization error estimator for the driven cavity cases outperforming richardson extrapolation which requires two grid levels however mnp has difficulties with the simpler heat conduction case due to the discretization errors having the same magnitude as the spline fitting errors

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

on the mean density of complex eigenvalues for an ensemble of random matrices with prescribed singular values given any fixed n n positive semi definite diagonal matrix g 0 we derive the explicit formula for the density of complex eigenvalues for random matrices a of the form where the random unitary matrices u are distributed on the group u n according to the haar measure

eigenvalue distributions of large hermitian matrices wigner s semi circle law and a theorem of kac murdock and szego abstract wigner s semi circle law describes the eigenvalue distribution of certain large random hermitian matrices a new proof is given for the case of gaussian matrices that involves reducing a random matrix to tridiagonal form by a method that is well known as a technique for numerical computation of eigenvalues the result is a generalized toeplitz matrix whose eigenvalue distribution can be found using a theorem of kac murdock and szego a new and more elementary proof of the latter is also given the arguments use only direct l 2 estimates rather than the transform methods or moment calculations employed previously

schur polynomials and biorthogonal random matrix ensembles the study of the average of schur polynomials over a stieltjes wigert ensemble has been carried out by dolivet and tierz j math phys 48 023507 2007 e print arxiv hep th 0609167 where it was shown that it is equal to quantum dimensions using the same approach we extend the result to the biorthogonal case we also study using the littlewood richardson rule some particular cases of the quantum dimension result finally we show that the notion of giambelli compatibility of schur averages introduced by borodin et al adv appl math 37 209 2006 e print arxiv math ph 0505021 also holds in the biorthogonal setting

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

autocorrelation of random matrix polynomials we calculate the autocorrelation functions or shifted moments of the characteristic polynomials of matrices drawn uniformly with respect to haar measure from the groups u n o 2n and usp 2n in each case the result can be expressed in three equivalent forms as a determinant sum and hence in terms of symmetric polynomials as a combinatorial sum and as a multiple contour integral these formulae are analogous to those previously obtained for the gaussian ensembles of random matrix theory but in this case are identities for any size of matrix rather than large matrix asymptotic approximations they also mirror exactly the autocorrelation formulae conjectured to hold for l functions in a companion paper this then provides further evidence in support of the connection between random matrix theory and the theory of l functions

generalizations of some integrals over the unitary group using the character expansion method we generalize several well known integrals over the unitary group to the case where general complex matrices appear in the integrand these integrals are of interest in the theory of random matrices and may also find applications in lattice gauge theory

pfaffian point process for the gaussian real generalised eigenvalue problem the generalised eigenvalues for a pair of n n matrices x1 x2 are defined as the solutions of the equation det x1 x2 0 or equivalently for x2 invertible as the eigenvalues of x 2 1 x 1 we consider gaussian real matrices x1 x2 for which the generalised eigenvalues have the rotational invariance of the half sphere or after a fractional linear transformation the rotational invariance of the unit disk in these latter variables we calculate the joint eigenvalue probability density function the probability pn k of finding k real eigenvalues the densities of real and complex eigenvalues the latter being related to an average over characteristic polynomials and give an explicit pfaffian formula for the higher correlation functions rho k 1 k 2 a limit theorem for pn k is proved and the scaled form of rho k 1 k 2 is shown to be identical to the analogous limit for the correlations of the eigenvalues of real gaussian matrices we show that these correlations satisfy sum rules characteristic of the underlying two component coulomb gas

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

brown s spectral distribution measure for r diagonal elements in finite von neumann algebras abstract in 1983 l g brown introduced a spectral distribution measure for non normal elements in a finite von neumann algebra m with respect to a fixed normal faithful tracial state in this paper we compute brown s spectral distribution measure in case t has a polar decomposition t uh where u is a haar unitary and u and h are free when ker t 0 this is equivalent to that t t is an r diagonal pair in the sense of nica and speicher the measure t is expressed explicitly in terms of the s transform of the distribution t t of the positive operator t t in case t is a circular element i e t x1 ix2 2 where x1 x2 is a free semicircular system then sp t d the closed unit disk and t has constant density 1 on d

the selberg jack symmetric functions abstract k aomoto has recently given a simple proof of an extension of a selberg s integral we prove the following generalization of aomoto s theorem for eachk 0 there exists a family sk t of homogeneous symmetric functions such that if the integrand of selberg s integral is multiplied bysk t1 tn then the integral has a certain closed form sincesk t is a renormalized jack symmetric function with 1 k we call sk t the selberg jack symmetric functions r p stanley and i g macdonald have extended many of the basic properties of the i j schur functions to the jack symmetric functions stanley s extensions of the pieri formula in a form obtained by macdonald s extension of the duality property and the combinatorial representation arise naturally in our proof we give proofs of these and other results of stanley and macdonald we give two constant term identities which are equivalent to the integration formula these results extend w g morris theorem foranand the macdonald morris constant term identity forbcn by exploring selberg s proof of his integral we give a recurrence relation outline a method for calculating the selberg jack symmetric functions and show that a constant term orthogonality relation recently proved by macdonald arises naturally we give an expansion which allows us to evaluate the normalization of this orthogonality thus giving an alternative proof of the integration formula

products and ratios of characteristic polynomials of random hermitian matrices we present new and streamlined proofs of various formulas for products and ratios of characteristic polynomials of random hermitian matrices that have appeared recently in the literature

integrals over the circular ensembles relating to classical domains using the theory of classical domains and jack polynomials we generalize several integrals over the unitary group due to fyodorov and khoruzhenko to integrals over the circular orthogonal and circular symplectic ensembles while in the unitary case a fundamental role is played by complex matrices it is anti symmetric and symmetric matrices which play the same role in the latter two cases in our workings a generalization of the evaluation of the integral of a jack polynomial times the selberg weight sought by fyodorov and khoruzhenko is identified

correlation functions of harish chandra integrals over the orthogonal and the symplectic groups the harish chandra correlation functions i e integrals over compact groups of invariant monomials prod operatorname tr x p 1 varomega y q 1 varomega dagger x p 2 cdots with the weight exp tr x y are computed for the orthogonal and symplectic groups we proceed in two steps first the integral over the compact group is recast into a gaussian integral over strictly upper triangular complex matrices with some additional symmetries supplemented by a summation over the weyl group this result follows from the study of loop equations in an associated two matrix integral and may be viewed as the adequate version of duistermaat heckman s theorem for our correlation function integrals secondly the gaussian integration over triangular matrices is carried out and leads to compact determinantal expressions

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