probability of all eigenvalues real for products of standard gaussian matrices with xi independent n n standard gaussian random matrices the probability that all eigenvalues are real for the matrix product pm xmxm 1 x1 is expressed in terms of an n 2 n 2 n even and n 1 2 n 1 2 n odd determinant the entries of the determinant are certain meijer g functions in the case m 2 high precision computation indicates that the entries are rational multiples of 2 with the denominator a power of 2 and that to leading order in n decays as we are able to show that for general m and large n with an explicit bm an analytic demonstration that as m is given

universal microscopic correlation functions for products of independent ginibre matrices we consider the product of n complex non hermitian independent random matrices each of size n n with independent identically distributed gaussian entries ginibre matrices the joint probability distribution of the complex eigenvalues of the product matrix is found to be given by a determinantal point process as in the case of a single ginibre matrix but with a more complicated weight given by a meijer g function depending on n using the method of orthogonal polynomials we compute all eigenvalue density correlation functions exactly for finite n and fixed n they are given by the determinant of the corresponding kernel which we construct explicitly in the large n limit at fixed n we first determine the microscopic correlation functions in the bulk and at the edge of the spectrum after unfolding they are identical to that of the ginibre ensemble with n 1 and thus universal in contrast the microscopic correlations we find at the origin differ for each n 1 and generalize the known bessel law in the complex plane for n 2 to a new hypergeometric kernel 0fn 1

entanglement optimizing mixtures of two qubit states entanglement in incoherent mixtures of pure states of two qubits is considered via the concurrence measure a set of pure states is optimal if the concurrence for any mixture of them is the weighted sum of the concurrences of the generating states when two or three pure real states are mixed it is shown that 28 5 or 5 12 of the cases respectively are optimal conditions that are obeyed by the pure states generating such optimally entangled mixtures are derived for four or more pure states it is shown that there are no such sets of real states the implication of these on the superposition of two or more dimerized states is discussed a corollary of these results also show in how many cases rebit concurrence can be the same as that of qubit concurrence

products of random matrices and generalised quantum point scatterers to every product of 2 2 matrices there corresponds a one dimensional schrodinger equation whose potential consists of generalised point scatterers products of random matrices are obtained by making these interactions and their positions random we exhibit a simple one dimensional quantum model corresponding to the most general product of matrices in sl 2 we use this correspondence to find new examples of products of random matrices for which the invariant measure can be expressed in simple analytical terms

minimum action method for the kardar parisi zhang equation we apply a numerical minimum action method derived from the wentzell freidlin theory of large deviations to the kardar parisi zhang equation for the height profile of a growing interface in one dimension we find that the transition pathway between different height configurations is determined by the nucleation and subsequent propagation of facets or steps corresponding to moving domain walls or growth modes in the underlying noise driven burgers equation this transition scenario is in accordance with recent analytical studies of the one dimensional kardar parisi zhang equation in the asymptotic weak noise limit we also briefly discuss transitions in two dimensions

singular value correlation functions for products of wishart random matrices we consider the product of m quadratic random matrices with complex elements and no further symmetry where all matrix elements of each factor have a gaussian distribution this generalizes the classical wishart laguerre gaussian unitary ensemble with m 1 in this paper we first compute the joint probability distribution for the singular values of the product matrix when the matrix size n and the number m are fixed but arbitrary this leads to a determinantal point process which can be realized in two different ways first it can be written as a one matrix singular value model with a nonstandard jacobian or second for m 2 as a two matrix singular value model with a set of auxiliary singular values and a weight proportional to the meijer g function for both formulations we determine all singular value correlation functions in terms of the kernels of biorthogonal polynomials which we explicitly construct they are given in terms of the hypergeometric and meijer g functions generalizing the laguerre polynomials for m 1 our investigation was motivated from applications in telecommunication of multi layered scattering multiple input and multiple output channels we present the ergodic mutual information for finite n for such a channel model with m 1 layers of scatterers as an example

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

products of rectangular random matrices singular values and progressive scattering we discuss the product of m rectangular random matrices with independent gaussian entries which have several applications including wireless telecommunication and econophysics for complex matrices an explicit expression for the joint probability density function is obtained using the harish chandra itzykson zuber integration formula explicit expressions for all correlation functions and moments for finite matrix sizes are obtained using a two matrix model and the method of biorthogonal polynomials this generalizes the classical result for the so called wishart laguerre gaussian unitary ensemble or chiral unitary ensemble at m 1 and previous results for the product of square matrices the correlation functions are given by a determinantal point process where the kernel can be expressed in terms of meijer g functions we compare the results with numerical simulations and known results for the macroscopic level density in the limit of large matrices the location of the end points of support for the latter are analyzed in detail for general m finally we consider the so called ergodic mutual information which gives an upper bound for the spectral efficiency of a mimo communication channel with multifold scattering

products of independent quaternion ginibre matrices and their correlation functions we discuss the product of independent induced quaternion 4 ginibre matrices and the eigenvalue correlations of this product matrix the joint probability density function for the eigenvalues of the product matrix is shown to be identical to that of a single ginibre matrix but with a more complicated weight function we find the skew orthogonal polynomials corresponding to the weight function of the product matrix and use the method of skew orthogonal polynomials to compute the eigenvalue correlation functions for product matrices of finite size the radial behavior of the density of states is studied in the limit of large matrices and the macroscopic density is discussed the microscopic limit at the origin at the edge s and in the bulk is also discussed for the radial behavior of the density of states

lyapunov exponents for products of rectangular real complex and quaternionic ginibre matrices we study the joint density of eigenvalues for products of independent rectangular real complex and quaternionic ginibre matrices in the limit where the number of matrices tends to infinity it is shown that the joint probability density function for the eigenvalues forms a permanental point process for all three classes the moduli of the eigenvalues become uncorrelated and log normal distributed while the distribution for the phases of the eigenvalues depends on whether real complex or quaternionic ginibre matrices are considered in the derivation for a product of real matrices we explicitly use the fact that all eigenvalues become real when the number of matrices tends to infinity finally we compare our results with known results for the lyapunov exponents as well as numerical simulations

kardar parisi zhang equation in the weak noise limit pattern formation and upper critical dimension we extend the previously developed nonperturbative weak noise scheme applied to the noisy burgers equation in one dimension to the kardar parisi zhang equation for a growing interface in arbitrary dimensions by means of the cole hopf transformation we show that the growth morphology can be interpreted in terms of dynamically evolving textures of localized growth modes with superimposed diffusive modes in the cole hopf representation the growth modes are static solutions to the diffusion equation and the nonlinear schrodinger equation subsequently boosted to finite velocity by a galilei transformation we discuss the dynamics of the pattern formation and briefly the superimposed linear modes implementing the stochastic interpretation we discuss kinetic transitions and in particular the preliminary scaling properties pertaining to the pair mode or dipole sector in the dipole sector we obtain the hurst exponent h 3 d 4 d or dynamic exponent zdip 4 d 3 d for the random walk of growth modes below d 3 the dipole growth modes show anomalous diffusion above d 3 the dipole growth modes freeze finally applying derrick s theorem based on constrained minimization we show that the upper critical dimension is d 4 in the sense that growth modes cease to exist above this dimension

anisotropic model of kinetic roughening the strong coupling regime we study the strong coupling sc limit of the anisotropic kardar parisi zhang kpz model a systematic mapping of the continuum model to its lattice equivalent shows that in the sc limit anisotropic perturbations destroy all spatial correlations but retain a temporal scaling which shows a remarkable crossover along one of the two spatial directions the choice of direction depending on the relative strength of anisotropicity the results agree with exact numerics and are expected to settle the long standing sc problem of a kpz model in the infinite range limit 2007 the american physical society

universal microscopic correlation functions for products of truncated unitary matrices we investigate the spectral properties of the product of m complex non hermitian random matrices that are obtained by removing l rows and columns of larger unitary random matrices uniformly distributed on the group u n l such matrices are called truncated unitary matrices or random contractions we first derive the joint probability distribution for the complex eigenvalues of the product matrix for fixed n l and m given by a standard determinantal point process in the complex plane the weight however is non standard and can be expressed in terms of the meijer g function the explicit knowledge of all eigenvalue correlation functions and the corresponding kernel allows us to take various large n and l limits at fixed m at strong non unitarity with l n finite the eigenvalues condense on a domain inside the unit circle at the edge and in the bulk we find the same universal microscopic kernel as for a single complex non hermitian matrix from the ginibre ensemble at the origin we find the same new universality classes labeled by m as for the product of m matrices from the ginibre ensemble keeping a fixed size of truncation l when n goes to infinity leads to weak non unitarity with most eigenvalues on the unit circle as for unitary matrices here we find a new microscopic edge kernel that generalizes the known results for m 1 we briefly comment on the case when each product matrix results from a truncation of different size lj

non hermitean wishart random matrices i a non hermitean extension of paradigmatic wishart random matrices is introduced to set up a theoretical framework for statistical analysis of real complex and real quaternion stochastic time series representing two remote complex systems the first paper in a series provides a detailed spectral theory of non hermitean wishart random matrices composed of complex valued entries the great emphasis is placed on an asymptotic analysis of the mean eigenvalue density for which we derive among other results a complex plane analog of the marcenko pastur law a surprising connection with a class of matrix models previously invented in the context of quantum chromodynamics is pointed out

on the number of real eigenvalues of products of random matrices and an application to quantum entanglement the probability that there are k real eigenvalues for an n dimensional real random matrix is known here we study this for the case of products of independent random matrices relating the problem of the probability that the product of two real two dimensional random matrices has real eigenvalues to an issue of optimal quantum entanglement this is fully analytically solved it is shown that in 4 fraction of such products the eigenvalues are real being greater than the corresponding known probability for a single matrix it is shown numerically that the probability that all eigenvalues of a product of random matrices are real tends to unity as the number of matrices in the product increases indefinitely some other numerical explorations including the expected number of real eigenvalues are also presented where an exponential approach of the expected number to the dimension of the matrix seems to hold

critical exponents of the kpz equation via multi surface coding numerical simulations we study the kpz equation in d 2 3 and 4 spatial dimensions by using a restricted solid on solid discretization of the surface we measure the critical exponents very precisely and we show that the rational guess is not appropriate and that d 4 is not the upper critical dimension we are also able to determine very precisely the exponent of the sub leading scaling corrections that turns out to be close to unity in all cases we introduce and use a multi surface coding technique that allows a gain of the order of 30 fold over usual numerical simulations

products of independent elliptic random matrices for fixed m 1 we study the product of m independent n times n elliptic random matrices as n tends to infinity our main result shows that the empirical spectral distribution of the product converges with probability 1 to the m th power of the circular law regardless of the joint distribution of the mirror entries in each matrix this leads to a new kind of universality phenomenon the limit law for the product of independent random matrices is independent of the limit laws for the individual matrices themselves our result also generalizes earlier results of gotze tikhomirov on the asymptotic spectrum of products of independent random matrices available at http arxiv org abs 1012 2710 and o rourke soshnikov j probab 16 81 2219 2245 2011 concerning the product of independent iid random matrices

newton s method for solving inclusions using set valued approximations results on stability of both local and global metric regularity under set valued perturbations are presented as an application we study super linear convergence of a newton type iterative process for solving generalized equations we investigate several iterative schemes such as the inexact newton s method the nonsmooth newton s method for semismooth functions the inexact proximal point algorithm etc moreover we also cover a forward backward splitting algorithm for finding a zero of the sum of two multivalued not necessarily monotone operators finally a globalization of the newton s method is discussed

the lyapunov exponent of products of random 2 2 matrices close to the identity we study products of arbitrary random real 2 2 matrices that are close to the identity matrix using the iwasawa decomposition of sl 2 we identify a continuum regime where the mean values and the covariances of the three iwasawa parameters are simultaneously small in this regime the lyapunov exponent of the product is shown to assume a scaling form in the general case the corresponding scaling function is expressed in terms of gauss hypergeometric function a number of particular cases are also considered where the scaling function of the lyapunov exponent involves other special functions airy bessel whittaker elliptic the general solution thus obtained allows us among other things to recover in a unified framework many results known previously from exactly solvable models of one dimensional disordered systems

diffusion of directed polymers in a random environment we consider a system of random walks or directed polymers interacting weakly with an environment which is random in space and time in spatial dimensionsd 2 we establish that the behavior is diffusive with probability one the diffusion constant is not renormalized by the interaction

real eigenvalue statistics for products of asymmetric real gaussian matrices abstract random matrices formed from i i d standard real gaussian entries have the feature that the expected number of real eigenvalues is non zero this property persists for products of such matrices independently chosen and moreover it is known that as the number of matrices in the product tends to infinity the probability that all eigenvalues are real tends to unity we quantify the distribution of the number of real eigenvalues for products of finite size real gaussian matrices by giving an explicit pfaffian formula for the probability that there are exactly k real eigenvalues as a determinant with entries involving particular meijer g functions we also compute the explicit form of the pfaffian correlation kernel for the correlation between real eigenvalues and the correlation between complex eigenvalues the simplest example of these the eigenvalue density of the real eigenvalues gives by integration the expected number of real eigenvalues our ability to perform these calculations relies on the construction of certain skew orthogonal polynomials in the complex plane the computation of which is carried out using their relationship to particular random matrix averages

stability of mann s iterates under metric regularity we consider a mann like iteration for solving the inclusion email protected t x where t email protected x is a set valued mapping defined from a banach space x into itself which is metrically regular near a point email protected email protected in its graph we study the behavior of the iterates generated by our method and prove that they inherit the regularity properties of the mapping t first we consider the case when the mapping t is metrically regular then the case when it is strongly metrically regular finally we present an inexact version of our method and we study its convergence when the mapping t is strongly metrically subregular

universal distribution of lyapunov exponents for products of ginibre matrices starting from exact analytical results on singular values and complex eigenvalues of products of independent gaussian complex random n n matrices also called the ginibre ensemble we rederive the lyapunov exponents for an infinite product we show that for a large number t of product matrices the distribution of each lyapunov exponent is normal and we compute its t dependent variance as well as corrections in a large t expansion originally lyapunov exponents are defined for the singular values of the product matrix that represents a linear time evolution surprisingly a similar construction for the moduli of the complex eigenvalues yields the very same exponents and normal distributions to leading order we discuss a general mechanism for 2 2 matrices showing why the singular values and the radii of complex eigenvalues collapse onto the same value in the large t limit we thereby rederive newman s triangular law which has a simple interpretation as the radial density of complex eigenvalues in the circular law and study the commutativity of the two limits and on the global scale and on the local scale we show as a mathematical byproduct that a particular asymptotic expansion of a meijer g function with large index leads to a gaussian

distribution of the determinant of a random real symmetric matrix from the gaussian orthogonal ensemble random matrix theory finds considerable use in various branches of physics notably in nuclear physics quantum chaology and for investigating hamiltonians of disordered and strongly interacting quantum systems 1 3 and references therein among the ensembles of n3n random matrices hn three gaussian ensembles have been studied extensively 1 3 and are still being investigated their probability densities are proportional to exp 2tr hn where tr means trace matrices are real symmetric for the gaussian orthogonal ensemble goe hermitian for the gaussian unitary ensemble gue and quaternion self dual for the gaussian symplectic ensemble many characteristics both exact at finite n and asymptotic at large n of the distributions of eigenvalues of n3n random matrices of various symmetries are known for the gaussian ensembles 1 3 the properties of eigenvalues of random matrix ensembles rme s can be interpreted in two dimensions 2d from the equilibrium characteristics of a gas of n identical point charges on a line 1 10 often referred to as a log gas 9 10 which interact via a logarithmic coulomb potential and are confined by an external potential the external potential is harmonic in the case of gaussian ensembles as can be seen by rewriting eq 1 below as a boltzmann factor at a temperature 1 b that depends only on the symmetry of the considered ensemble with b51 2 and 4 for the orthogonal unitary and symplectic ensembles respectively 1 3 the problems of the distribution and associated fluctuations of linear statistics in rme s 1 10 and references therein are of interest for instance in the study of conductance fluctuations in mesoscopic conductors the linear statistics problem deals with physical quantities f which are given by sums f5 k51 n f l k over the eigenvalues l k of a random matrix where f x may depend nonlinearly on x general arguments have been used to predict that the distribution of any linear statistic is gaussian and independent of the confining potential in the scaled asymptotic limit n provided its variance is finite 5 10 the potential fluctuations have been investigated in 2d and 3d classical charged systems 11 12 and more recently for circular ensembles by baker and forrester who further studied the dipole moment statistic 9 within the log gas interpretation the statistic of the potential at the origin is obtained for f x 52lnuxu that is for f5 2lnudu it may thus be derived directly from the distribution of the determinant d of hn we notice further that the partition functions of log gases whose confining harmonic potential includes a supplementary logarithmic contribution are by products of the derivation of the determinant distributions of gaussian ensembles see eqs 10 12 19 and 26 for b51 the rme s associated with the latter log gases belong to the family of generalized gaussian ensembles 13 14

eigenvalue statistics of the real ginibre ensemble the real ginibre ensemble consists of random n ifmmode times else texttimes fi n matrices formed from independent and identically distributed standard gaussian entries by using the method of skew orthogonal polynomials the general n point correlations for the real eigenvalues and for the complex eigenvalues are given as n ifmmode times else texttimes fi n pfaffians with explicit entries a computationally tractable formula for the cumulative probability density of the largest real eigenvalue is presented this is relevant to may s stability analysis of biological webs

the distribution and moments of the smallest eigenvalue of a random matrix of wishart type abstract given a random rectangular m n matrix with elements from a normal distribution what is the distribution of the smallest singular value to pose an equivalent question in the language of multivariate statistics what is the distribution of the smallest eigenvalue of a matrix from the central wishart distribution in the null case we give new results giving the distribution as a simple recursion this includes the more difficult case when n m is an even integer without resorting to zonal polynomials and hypergeometric functions of matrix arguments with the recursion one can obtain exact expressions for the density and the moments of the distribution in terms of functions usually no more complicated than polynomials exponentials and at worst ordinary hypergeometric functions we further elaborate on the special cases when n m 0 1 2 and 3 and give a numerical table of the expected values for 2 m 25 and 0 n m 25

integrals of the ising class from an experimental mathematical perspective we analyze ising class integrals our experimental results involvedextreme precision multidimensional quadrature on intricate integrands thus highly parallel computation was required

spectral statistics for the difference of two wishart matrices in this work we consider the weighted difference of two independent complex wishart matrices and derive the joint probability density function of the corresponding eigenvalues in a finite dimension scenario using two distinct approaches the first derivation involves the use of unitary group integral while the second one relies on applying the derivative principle the latter relates the joint probability density of eigenvalues of a matrix drawn from a unitarily invariant ensemble to the joint probability density of its diagonal elements exact closed form expressions for an arbitrary order correlation function are also obtained and spectral densities are contrasted with monte carlo simulation results analytical results for moments as well as probabilities quantifying positivity aspects of the spectrum are also derived additionally we provide a large dimension asymptotic result for the spectral density using the stieltjes transform approach for algebraic random matrices finally we point out the relationship of these results with the corresponding results for difference of two random density matrices and obtain some explicit and closed form expressions for the spectral density and absolute mean

towards a strong coupling theory for the kpz equation abstract after a brief introduction we review the nonperturbative weak noise approach to the kardar parisi zhang kpz equation in one dimension we argue that the strong coupling aspects of the kpz equation are related to the existence of localized propagating domain walls or solitons representing the growth modes the statistical weight of the excitations is governed by a dynamical action representing the generalization of the boltzmann factor to kinetics this picture is not limited to one dimension we thus attempt a generalization to higher dimensions where the strong coupling aspects presumably are associated with a cellular network of domain walls based on this picture we present arguments for the wolf kertez expression z 2d 1 d 1 for the dynamical exponent

the cauchy two matrix model we introduce a new class of two multi matrix models of positive hermitian matrices coupled in a chain the coupling is related to the cauchy kernel and differs from the exponential coupling more commonly used in similar models the correlation functions are expressed entirely in terms of certain biorthogonal polynomials and solutions of appropriate riemann hilbert problems thus paving the way to a steepest descent analysis and universality results the interpretation of the formal expansion of the partition function in terms of multicolored ribbon graphs is provided and a connection to the o 1 model a steepest descent analysis of the partition function reveals that the model is related to a trigonal curve three sheeted covering of the plane much in the same way as the hermitian matrix model is related to a hyperelliptic curve

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

functional renormalization group and the field theory of disordered elastic systems we study elastic systems such as interfaces or lattices pinned by quenched disorder to escape triviality as a result of dimensional reduction we use the functional renormalization group difficulties arise in the calculation of the renormalization group functions beyond one loop order even worse observables such as the two point correlation function exhibit the same problem already at one loop order these difficulties are due to the nonanalyticity of the renormalized disorder correlator at zero temperature which is inherent to the physics beyond the larkin length characterized by many metastable states as a result two loop diagrams which involve derivatives of the disorder correlator at the nonanalytic point are naively ambiguous we examine several routes out of this dilemma which lead to a unique renormalizable field theory at two loop order it is also the only theory consistent with the potentiality of the problem the beta function differs from previous work and the one at depinning by novel anomalous terms for interfaces and random bond disorder we find a roughness exponent zeta 0 208 298 04epsilon 0 006 858epsilon 2 epsilon 4 d for random field disorder we find zeta epsilon 3 and compute universal amplitudes to order o epsilon 2 for periodic systems we evaluate the universal amplitude of the two point function we also clarify the dependence of universal amplitudes on the boundary conditions at large scale all predictions are in good agreement with numerical and exact results and are an improvement over one loop finally we calculate higher correlation functions which turn out to be equivalent to those at depinning to leading order in epsilon

ishikawa and mann iteration methods with errors for nonlinear equations of the accretive type abstract let e be an arbitrary banach space and t e e a lipschitz strongly accretive operator it is proved that for a given f e the ishikawa and the mann iteration methods with errors introduced by 18 converge strongly to the solution of the equation tx f furthermore if e is a uniformly smooth banach space and t e e is demicontinuous and strongly accretive it is also proved that both the ishikawa and the mann iteration methods with errors converge strongly to the solution of the equation tx f related results deal with the iterative approximation of fixed points of strongly pseudocontractive operators and the solution of the equation x tx f f e when t e e is m accretive

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

product of ginibre matrices fuss catalan and raney distributions squared singular values of a product of s square random ginibre matrices are asymptotically characterized by probability distributions p s x such that their moments are equal to the fuss catalan numbers of order s we find a representation of the fuss catalan distributions p s x in terms of a combination of s hypergeometric functions of the type s f s 1 the explicit formula derived here is exact for an arbitrary positive integer s and for s 1 it reduces to the marchenko pastur distribution using similar techniques involving the mellin transform and the meijer g function we find exact expressions for the raney probability distributions the moments of which are given by a two parameter generalization of the fuss catalan numbers these distributions can also be considered as a two parameter generalization of the wigner semicircle law

exact evaluations of some meijer g functions and probability of all eigenvalues real for the product of two gaussian matrices we provide a proof to a recent conjecture by forrester 2014 j phys a math theor 47 065202 regarding the algebraic and arithmetic structure of meijer g functions which appear in the expression for probability of all eigenvalues real for the product of two real gaussian matrices in the process we come across several interesting identities involving meijer g functions

statistics of real eigenvalues in ginibre s ensemble of random real matrices the integrable structure of ginibre s orthogonal ensemble of random matrices is looked at through the prism of the probability p n k to find exactly k real eigenvalues in the spectrum of an n ifmmode times else texttimes fi n real asymmetric gaussian random matrix the exact solution for the probability function p n k is presented and its remarkable connection to the theory of symmetric functions is revealed an extension of the dyson integration theorem is a key ingredient of the theory presented

probability density of determinants of random matrices in this brief paper the probability density of a random real complex and quaternion determinant is rederived using the singular values the behaviour of suitably rescaled random determinants is studied in the limit of infinite order of the matrices

asymptotics of finite system lyapunov exponents for some random matrix ensembles for products pn of n random matrices of size there is a natural notion of finite n lyapunov exponents in the case of standard gaussian random matrices with real complex or real quaternion elements and extended to the general variance case for methods known for the computation of are used to compute the large n form of the variances of the exponents analogous calculations are performed in the case that the matrices making up pn are products of sub blocks of random unitary matrices with haar measure furthermore we make some remarks relating to the coincidence of the lyapunov exponents and the stability exponents relating to the eigenvalues of pn

on the perturbation expansion of the kpz equation we present a simple argument to show that the function of the d dimensional kpz equation d 2 is to all orders in perturbation theory given by beta g r d 2 g r 2 8 pi d 2 text gamma 2 d 2 g r 2 neither the dynamical exponent z nor the roughness exponent have any correction in any order of perturbation theory this shows that standard perturbation theory cannot attain the strong coupling regime and in addition breaks down at d 4 we also calculate a class of correlation functions exactly

eigenvalues and singular values of products of rectangular gaussian random matrices we derive exact analytic expressions for the distributions of eigenvalues and singular values for the product of an arbitrary number of independent rectangular gaussian random matrices in the limit of large matrix dimensions we show that they both have power law behavior at zero and determine the corresponding powers we also propose a heuristic form of finite size corrections to these expressions which very well approximates the distributions for matrices of finite dimensions

spectrum of the product of independent random gaussian matrices we show that the eigenvalue density of a product x x1x2 xm of m independent nxn gaussian random matrices in the limit n infinity is rotationally symmetric in the complex plane and is given by a simple expression rho z z 1 mpisigma 2 m z 2 2 m for z sigma the parameter sigma corresponds to the radius of the circular support and is related to the amplitude of the gaussian fluctuations this form of the eigenvalue density is highly universal it is identical for products of gaussian hermitian non hermitian and real or complex random matrices it does not change even if the matrices in the product are taken from different gaussian ensembles we present a self contained derivation of this result using a planar diagrammatic technique additionally we conjecture that this distribution also holds for any matrices whose elements are independent centered random variables with a finite variance or even more generally for matrices which fulfill pastur lindeberg s condition we provide a numerical evidence supporting this conjecture

critical discussion of the two loop calculations for the kardar parisi zhang equation in this article we perform a careful analysis of the renormalization procedure used in existing calculations to derive critical exponents for the kpz equation at 2 loop order this analysis explains the discrepancies between the results of the different groups the correct critical exponents in d 2 epsilon dimensions at the crossover between weak and strong coupling regime are zeta o epsilon 3 and z 2 o epsilon 3 no strong coupling fixed point exists at 2 loop order

eigenvector statistics of the product of ginibre matrices we develop a method to calculate left right eigenvector correlations of the product of m independent n n complex ginibre matrices for illustration we present explicit analytical results for the vector overlap for a couple of examples for small m and n we conjecture that the integrated overlap between left and right eigenvectors is given by the formula o 1 m 2 n 1 and support this conjecture by analytical and numerical calculations we derive an analytical expression for the limiting correlation density as n for the product of ginibre matrices as well as for the product of elliptic matrices in the latter case we find that the correlation function is independent of the eccentricities of the elliptic laws

generalized mann iterates for constructing fixed points in hilbert spaces the mean iteration scheme originally proposed by mann is extended to a broad class of relaxed inexact fixed point algorithms in hilbert spaces weak and strong convergence results are established under general conditions on the underlying averaging process and the type of operators involved this analysis significantly widens the range of applications of mean iteration methods several examples are given