on the characteristic polynomial of a random unitary matrix we present a range of fluctuation and large deviations results for the logarithm of the characteristic polynomial z of a random n n unitary matrix as n first we show that evaluated at a finite set of distinct points is asymptotically a collection of i i d complex normal random variables this leads to a refinement of a recent central limit theorem due to keating and snaith and also explains the covariance structure of the eigenvalue counting function next we obtain a central limit theorem for ln z in a sobolev space of generalised functions on the unit circle in this limiting regime lower order terms which reflect the global covariance structure are no longer negligible and feature in the covariance structure of the limiting gaussian measure large deviations results for ln z a evaluated at a finite set of distinct points can be obtained for for higher order scalings we obtain large deviations results for ln z a evaluated at a single point there is a phase transition at a ln n which only applies to negative deviations of the real part reflecting a switch from global to local conspiracy

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

correlation functions cluster functions and spacing distributions for random matrices the usual formulas for the correlation functions in orthogonal and symplectic matrix models express them as quaternion determinants from this representation one can deduce formulas for spacing probabilities in terms of fredholm determinants of matrix valued kernels the derivations of the various formulas are somewhat involved in this article we present a direct approach which leads immediately to scalar kernels for the unitary ensembles and matrix kernels for the orthogonal and symplectic ensembles and the representations of the correlation functions cluster functions and spacing distributions in terms of them

negative moments of characteristic polynomials of random goe matrices and singularity dominated strong fluctuations we calculate the negative integer moments of the regularized characteristic polynomials of n n random matrices taken from the gaussian orthogonal ensemble goe in the limit as n the results agree nontrivially with a recent conjecture of berry and keating motivated by techniques developed in the theory of singularity dominated strong fluctuations this is the first example where nontrivial predictions obtained using these techniques have been proved

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

characteristic polynomials of random matrices at edge singularities we have discussed earlier the correlation functions of the random variables mathrm det ensuremath lambda ensuremath x in which x is a random matrix in particular the moments of the distribution of these random variables are universal functions when measured in the appropriate units of the level spacing when the ensuremath lambda mathrm s instead of belonging to the bulk of the spectrum approach the edge a crossover takes place to an airy or to a bessel problem and we consider here these modified classes of universality furthermore when an external matrix source is added to the probability distribution of x various different phenomenona may occur and one can tune the spectrum of this source matrix to other critical points again there are remarkably simple formulas for arbitrary source matrices which allow us to compute the moments of the characteristic polynomials in these cases as well

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

non intersecting paths random tilings and random matrices we investigate certain measures induced by families of non intersecting paths in domino tilings of the aztec diamond rhombus tilings of an abc hexagon a dimer model on a cylindrical brick lattice and a growth model the measures obtained e g the krawtchouk and hahn ensembles have the same structure as the eigenvalue measures in random matrix theory like gue which can in fact can be obtained from non intersecting brownian motions the derivations of the measures are based on the karlin mcgregor or lindstrom gessel viennot method we use the measures to show some asymptotic results for the models

riemannian symmetric superspaces and their origin in random matrix theory gaussian random matrix ensembles defined over the tangent spaces of the large families of cartan s symmetric spaces are considered such ensembles play a central role in mesoscopic physics as they describe the universal ergodic limit of disordered and chaotic single particle systems the generating function for the spectral correlations of each ensemble is reduced to an integral over a riemannian symmetric superspace in the limit of large matrix dimension such a space is defined as a pair g h m r where g h is a complex analytic graded manifold homogeneous with respect to the action of a complex lie supergroup g and m r is a maximal riemannian submanifold of the support of g h

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

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

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

correlation kernels for discrete symplectic and orthogonal ensembles in 49 h widom derived formulae expressing correlation functions of orthogonal and symplectic ensembles of random matrices in terms of orthogonal polynomials we obtain similar results for discrete ensembles with rational discrete logarithmic derivative and compute explicitly correlation kernels associated to the classical meixner and charlier orthogonal polynomials

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

janossy densities ii pfaffian ensembles we extend the main result of the companion paper j stat phys 113 595 610 to the case of the pfaffian ensembles

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

universality of the second mixed moment of the characteristic polynomials of the 1d band matrices real symmetric case we prove that the asymptotic behavior of the second mixed moment of the characteristic polynomials of the n n 1d gaussian real symmetric band matrices with the width of the band w n1 2 coincides with those for the gaussian orthogonal ensemble here we adapt the approach of shcherbina commun math phys 328 45 82 2014 where the hermitian case was considered

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

new correlation functions for random matrices and integrals over supergroups the averages of ratios of characteristic polynomials det x of n n random matrices x are investigated in the large n limit for the gue goe and gse ensembles the density of states and the two point correlation function are derived from these ratios the method relies on an extension of the harish chandra itzykson zuber integrals to the goe ensemble and to supergroups which are explicitly computed through heat kernel differential equations an external matrix source linearly coupled to the random matrices may also be added to the gaussian distribution and allows for a discussion of the universality of the goe results in the large n limit

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

a note on the internal consistency of various preference representations in chiclana et al fuzzy sets and systems 97 1998 33 we presented a fuzzy multipurpose decision making model integrating different preference representations preference orderings utility functions and fuzzy preference relations we complete the decision model studying its internal consistency

on the relation between orthogonal symplectic and unitary matrix ensembles for the unitary ensembles of n n hermitian matrices associated with a weight function w there is a kernel expressible in terms of the polynomials orthogonal with respect to the weight function which plays an important role for the orthogonal and symplectic ensembles of hermitian matrices there are 2 2 matrix kernels usually constructed using skew orthogonal polynomials which play an analogous role these matrix kernels are determined by their upper left hand entries we derive formulas expressing these entries in terms of the scalar kernel for the corresponding unitary ensembles we also show that whenever w w is a rational function the entries are equal to the scalar kernel plus some extra terms whose number equals the order of w w general formulas are obtained for these extra terms we do not use skew orthogonal polynomials in the derivations

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

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

classical skew orthogonal polynomials and random matrices skew orthogonal polynomials arise in the calculation of the n point distribution function for the eigenvalues of ensembles of random matrices with orthogonal or symplectic symmetry in particular the distribution functions are completely determined by a certain sum involving the skew orthogonal polynomials in the case that the eigenvalue probability density function involves a classical weight function explicit formulas for the skew orthogonal polynomials are given in terms of related orthogonal polynomials and the structure is used to give a closed form expression for the sum this theory treates all classical cases on an equal footing giving formulas applicable at once to the hermite laguerre and jacobi cases

characteristic polynomials of real symmetric random matrices the correlation functions of the random variables det x in which x is an hermitian n n random matrix are known to exhibit universal local statistics in the large n limit we study here the correlation of those same random variables for real symmetric matrices goe the derivation relies on an exact dual representation of the problem the k point functions are expressed in terms of finite integrals over quaternionic k k matrices however the control of the dyson limit in which the distance of the various parameters s is of the order of the mean spacing requires an integration over the symplectic group it is shown that a generalization of the itzykson zuber method holds for this problem but contrary to the unitary case the semi classical result requires a finite number of corrections to be exact we have also considered the problem of an external matrix source coupled to the random matrix and obtain explicit integral formulae which are useful for the analysis of the large n limit

algebraic aspects of increasing subsequences we present a number of results relating partial cauchy littlewood sums integrals over the compact classical groups and increasing subsequences of permutations these include integral formulae for the distribution of the longest increasing subsequence of a random involution with constrained number of fixed points new formulae for partial cauchy littlewood sums as well as new proofs of old formulae relations of these expressions to orthogonal polynomials on the unit circle and explicit bases for invariant spaces of the classical groups together with appropriate generalizations of the straightening algorithm

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

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

duality of orthogonal polynomials on a finite set we prove a certain duality relation for orthogonal polynomials defined on a finite set the result is used in a direct proof of the equivalence of two different ways using particles or holes of computing the correlation functions of a discrete orthogonal polynomial ensemble

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

singularity dominated strong fluctuations for some random matrix averages the circular and jacobi ensembles of random matrices have their eigenvalue support on the unit circle of the complex plane and the interval 0 1 of the real line respectively the averaged value of the modulus of the corresponding characteristic polynomial raised to the power 2 diverges for 2 1 at points approaching the eigenvalue support using the theory of generalized hypergeometric functions based on jack polynomials the functional form of the leading asymptotic behaviour is established rigorously in the circular ensemble case this confirms a conjecture of berry and keating

matrix kernels for measures on partitions we consider the problem of computation of the correlation functions for the z measures with the deformation jack parameters 2 or 1 2 such measures on partitions are originated from the representation theory of the infinite symmetric group and in many ways are similar to the ensembles of random matrix theory of 4 or 1 symmetry types for a certain class of such measures we show that correlation functions can be represented as pfaffians including 2 2 matrix valued kernels and compute these kernels explicitly we also give contour integral representations for correlation kernels of closely connected measures on partitions

distributions on partitions point processes and the hypergeometric kernel we study a 3 parametric family of stochastic point processes on the one dimensional lattice originated from a remarkable family of representations of the infinite symmetric group we prove that the correlation functions of the processes are given by determinantal formulas with a certain kernel the kernel can be expressed through the gauss hypergeometric function we call it the hypergeometric kernel in a scaling limit our processes approximate the processes describing the decomposition of representations mentioned above into irreducibles as we showed in previous works the correlation functions of these limit processes also have determinantal form with so called whittaker kernel we show that the scaling limit of the hypergeometric kernel is the whittaker kernel integrable operator as defined by its izergin korepin and slavnov we argue that the hypergeometric kernel can be considered as a kernel defining a discrete integrable operator we also show that the hypergeometric kernel degenerates for certain values of parameters to the christoffel darboux kernel for meixner orthogonal polynomials this fact is parallel to the degeneration of the whittaker kernel to the christoffel darboux kernel for laguerre polynomials

weak commutation relations and eigenvalue statistics for products of rectangular random matrices we study the joint probability density of the eigenvalues of a product of rectangular real complex or quaternion random matrices in a unified way the random matrices are distributed according to arbitrary probability densities whose only restriction is the invariance under left and right multiplication by orthogonal unitary or unitary symplectic matrices respectively we show that a product of rectangular matrices is statistically equivalent to a product of square matrices hereby we prove a weak commutation relation of the random matrices at finite matrix sizes which previously has been discussed for infinite matrix size moreover we derive the joint probability densities of the eigenvalues to illustrate our results we apply them to a product of random matrices drawn from ginibre ensembles and jacobi ensembles as well as a mixed version thereof for these weights we show that the product of complex random matrices yields a determinantal point process while the real and quaternion matrix ensembles correspond to pfaffian point processes our results are visualized by numerical simulations furthermore we present an application to a transport on a closed disordered chain coupled to a particle bath

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

the supersymmetry method for chiral random matrix theory with arbitrary rotation invariant weights in the past few years the supersymmetry method has been generalized to real symmetric hermitian and hermitian self dual random matrices drawn from ensembles invariant under the orthogonal unitary and unitary symplectic groups respectively we extend this supersymmetry approach to chiral random matrix theory invariant under the three chiral unitary groups in a unifying way thereby we generalize a projection formula providing a direct link and hence a short cut between the probability density in ordinary space and that in superspace we emphasize that this point was one of the main problems and shortcomings of the supersymmetry method since only implicit dualities between ordinary space and superspace were known before to provide examples we apply this approach to the calculation of the supersymmetric analogue of a lorentzian cauchy ensemble and an ensemble with a quartic potential moreover we consider the partially quenched partition function of the three chiral gaussian ensembles corresponding to four dimensional continuum quantum chromodynamics we identify a natural splitting of the chiral lagrangian in its lowest order into a part for the physical mesons and a part associated with source terms generating the observables e g the level density of the dirac operator

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

the k point random matrix kernels obtained from one point supermatrix models the k point correlation functions of the gaussian random matrix ensembles are certain determinants of functions which depend on only two arguments they are referred to as kernels since they are the building blocks of all correlations we show that the kernels are obtained for arbitrary level number directly from supermatrix models for one point functions more precisely the generating functions of the one point functions are equivalent to the kernels this is surprising because it implies that already the one point generating function holds essential information about the k point correlations this also establishes a link to the averaged ratios of spectral determinants i e of characteristic polynomials

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

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

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

derivation and application of extended parabolic wave theories i the factorized helmholtz equation the reduced scalar helmholtz equation for a transversely inhomogeneous half space supplemented with an outgoing radiation condition and an appropriate boundary condition on the initial value plane defines a direct acoustic propagation model this elliptic formulation admits a factorization and is subsequently equivalent to a first order weyl pseudodifferential equation which is recognized as an extended parabolic propagation model perturbation treatments of the appropriate weyl composition equation result in a systematic development of approximate wave theories which extend the narrow angle weak inhomogeneity and weak gradient ordinary parabolic schrodinger approximation the analysis further provides for the formulation and exact solution of a multidimensional nonlinear inverse problem appropriate for ocean acoustic and seismic studies the wave theories foreshadow computational algorithms the inclusion of range dependent effects and the extension to 1 the vector formulation appropriate for elas

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

asymptotics of plancherel measures for symmetric groups 1 1 plancherel measures given a finite group g by the corresponding plancherel measure we mean the probability measure on the set g of irreducible representations of g which assigns to a representation g the weight dim g for the symmetric group s n the set s n is the set of partitions of the number n which we shall identify with young diagrams with n squares throughout this paper the plancherel measure on partitions arises naturally in representation theoretic combinatorial and probabilistic problems for example the plancherel distribution of the first part of a partition coincides with the distribution of the longest increasing subsequence of a uniformly distributed random permutation 31 we denote the plancherel measure on partitions of n by mn

replica symmetric approach to the typical eigenvalue fluctuations of gaussian random matrices we discuss an approach to compute the first and second moments of the number of eigenvalues i n that lie in an arbitrary interval of the real line for gaussian random matrices the method combines the standard replica symmetric theory with a perturbative expansion of the saddle point action up to leading to the correct logarithmic scaling of the variance as well as to an analytical expression for the correction to the average standard results for the number variance at the local scaling regime are recovered in the limit of a vanishing interval the limitations of the replica symmetric method are unveiled by comparing our results with those derived through exact methods the present work represents an important step to study the fluctuations of i n in non invariant random matrix ensembles where the joint distribution of eigenvalues is not known