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

products of random matrices we derive analytic expressions for infinite products of random 2 x 2 matrices the determinant of the target matrix is log normally distributed whereas the remainder is a surprisingly complicated function of a parameter characterizing the norm of the matrix and a parameter characterizing its skewness the distribution may have importance as an uncommitted prior in statistical image analysis

on the distribution of a random variable occurring in 1d disordered systems the authors consider the random variable z 1 x1 x1x2 x1x2x3 where the xi are independent identically distributed variables they derive some asymptotic properties of the distribution of z which are related e g to the low temperature behaviour of the random field ising chain for a special class of distributions of the xi exact solutions are presented they also study the cases where the distribution function of z exhibits a power law fall off modulated by a periodic critical amplitude

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

exactly soluble random field ising models in one dimension the authors solve exactly the one dimensional random field ising model for two classes of magnetic field distributions symmetric exponential model i and non symmetric exponential model ii for both models expressions for the free energy at all finite temperatures are presented the low temperature region is examined in more detail they obtain the zero temperature energy and entropy in closed form it is shown that the free energy of both models has an expansion in integer powers of temperature model i has a non vanishing zero point entropy for all values of the parameters as soon as randomness is diluted in model ii the zero point entropy is zero except for a discrete sequence of values of one parameter in some cases the zero temperature magnetisation is positive whereas the average magnetic field is negative the magnetisation may also change sign as a function of temperature

multiplying unitary random matrices universality and spectral properties in this paper we calculate in the large n limit the eigenvalue density of an infinite product of random unitary matrices each of them generated by a random hermitian matrix this is equivalent to solving the unitary diffusion generated by a hamiltonian random in time we find that the average eigenvalue density is universal and depends only on the second moment of the generator of the stochastic evolution we find indications of critical behaviour eigenvalue spacing scaling like 1 n3 4 close to for a specific critical evolution time tc

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

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

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

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

fluctuations of the product of random matrices and generalized lyapunov exponent i present a general framework allowing to carry out explicit calculation of the moment generating function of random matrix products varpi n m nm n 1 ldots m 1 where m i s are i i d following tutubalin theor probab appl 10 1 15 27 1965 the calculation of the generating function is reduced to finding the largest eigenvalue of a certain transfer operator associated with a family of representations of the group the formalism is illustrated by considering products of random matrices from the group mathrm sl 2 mathbb r where explicit calculations are possible for concreteness i study in detail transfer matrix products for the one dimensional schrodinger equation where the random potential is a levy noise derivative of a levy process in this case i obtain a general formula for the variance of ln varpi n and for the variance of ln psi x where psi x is the wavefunction in terms of a single integral involving the fourier transform of the invariant density of the matrix product finally i discuss the continuum limit of random matrix products matrices close to the identity in particular i investigate a simple case where the spectral problem providing the generalized lyapunov exponent can be solved exactly

complete blow up after tmax for the solution of a semilinear heat equation soit un sous ensemble ouvert borne de r n a frontiere lisse on considere le probleme u t u f u dans o t u 0 sur o t u x o u 0 x x ou f r r est localement de lipschitz non decroissante et f o 0

products of many large random matrices and gradients in deep neural networks we study products of random matrices in the regime where the number of terms and the size of the matrices simultaneously tend to infinity our main theorem is that the logarithm of the ell 2 norm of such a product applied to any fixed vector is asymptotically gaussian the fluctuations we find can be thought of as a finite temperature correction to the limit in which first the size and then the number of matrices tend to infinity depending on the scaling limit considered the mean and variance of the limiting gaussian depend only on either the first two or the first four moments of the measure from which matrix entries are drawn we also obtain explicit error bounds on the moments of the norm and the kolmogorov smirnov distance to a gaussian finally we apply our result to obtain precise information about the stability of gradients in randomly initialized deep neural networks with relu activations this provides a quantitative measure of the extent to which the exploding and vanishing gradient problem occurs in a fully connected neural network with relu activations and a given architecture

the triangle law for lyapunov exponents of large random matrices for products a t a t 1 a 1 of i i d n n random matrices with i i d entries a triangle law governs then distribution of lyapunov exponents much like wigner s quarter circle law governs the singular values ofa 1 our proof requires finite fourth moments and a bounded density the result was previously derived only in the gaussian case

global decaying solutions of a focusing energy critical heat equation in r4 abstract we study solutions of the focusing energy critical nonlinear heat equation u t u u 2 u in r 4 we show that solutions emanating from initial data with energy and h 1 norm below those of the stationary solution w are global and decay to zero via the concentration compactness plus rigidity strategy of kenig merle 33 34 first global such solutions are shown to dissipate to zero using a refinement of the small data theory and the l 2 dissipation relation finite time blow up is then ruled out using the backwards uniqueness of escauriaza seregin sverak 17 18 in an argument similar to that of kenig koch 32 for the navier stokes equations

construction of solutions with exactly k blow up points for the schrodinger equation with critical nonlinearity we consider the nonlinear schrodinger equation i partial u mathord left vphantom i partial u partial t right kern nulldelimiterspace partial t delta u left u right 4 mathord left vphantom 4 n right kern nulldelimiterspace n uandu left 0 right varphi left right 1 whereu 0 t n for any given pointsx1 x2 x k in n we construct a solution of eq 1 u t which blows up in a finite timet at exactlyx1 x2 x k in addition we describe the precise behavior of the solutionu t whent t at the blow up points x1 x2 x k and in n x1 x2 x k

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

spectral relations between products and powers of isotropic random matrices we show that the limiting eigenvalue density of the product of n identically distributed random matrices from an isotropic unitary ensemble is equal to the eigenvalue density of n th power of a single matrix from this ensemble in the limit when the size of the matrix tends to infinity using this observation one can derive the limiting density of the product of n independent identically distributed non hermitian matrices with unitary invariant measures in this paper we discuss two examples the product of n girko ginibre matrices and the product of n truncated unitary matrices we also provide evidence that the result holds also for isotropic orthogonal ensembles

explicit invariant measures for products of random matrices we construct explicit invariant measures for a family of infinite products of random independent identically distributed elements of sl 2 c the matrices in the product are such that one entry is gamma distributed along a ray in the complex plane when the ray is the positive real axis the products are those associated with a continued fraction studied by letac seshadri z wahr verw geb 62 1983 485 489 who showed that the distribution of the continued fraction is a generalised inverse gaussian we extend this result by finding the distribution for an arbitrary ray in the complex right half plane and thus compute the corresponding lyapunov exponent explicitly when the ray lies on the imaginary axis the matrices in the infinite product coincide with the transfer matrices associated with a one dimensional discrete schrodinger operator with a random gamma distributed potential hence the explicit knowledge of the lyapunov exponent may be used to estimate the exponential rate of localisation of the eigenstates

singular behaviour of certain infinite products of random 2 2 matrices the authors consider an infinite product of random matrices which appears in several physical problems in particular the ising chain in a random field the random matrices depend analytically on a parameter epsilon in such a way that for 0 they all commute under certain conditions they find that the lyapunov index of this product behaves approximately as c 2 for to 0 their approach is based on the decomposition of an exact integral equation for this problem into two reduced equations they give an expression for the exponent alpha in terms of the probability distribution of the matrices and for the proportionality constant c in terms of the solutions of the reduced integral equations in cases where exact results are available agreement is obtained the physical consequences for disordered one dimensional systems are pointed out

the distribution of the reflection phase of disordered conductors the reflection coefficients of one dimensional disordered conductors is asymptotically a pure phase when the sample length is much larger than the localisation length the authors study the stationary distribution of this reflection phase for tight binding chains with diagonal disorder i e random site potentials they derive the dyson integral equation obeyed by this distribution and relate it to the lyapunov exponent and the integrated density of states this general approach allows them to consider then more specifically two types of disorder in the case of a binary alloy where the potentials take two values the distribution of the reflection phase is shown to have generically an infinity of power law singularities related to the occurrence of periodic patterns of potentials this phenomenon produces divergences in the distribution when the strength of disorder exceeds some energy dependent threshold they also obtain an exact expression for the distribution of the reflection phase for a symmetric exponential distribution of potentials this non trivial solvable model allows them to examine analytically various properties of the phase distribution such as the anomalies which occur at weak disorder

wigner surmise for hermitian and non hermitian chiral random matrices we use the idea of a wigner surmise to compute approximate distributions of the first eigenvalue in chiral random matrix theory for both real and complex eigenvalues testing against known results for zero and maximal non hermiticity in the microscopic large n limit we find an excellent agreement valid for a small number of exact zero eigenvalues compact expressions are derived for real eigenvalues in the orthogonal and symplectic classes and at intermediate non hermiticity for the unitary and symplectic classes such individual dirac eigenvalue distributions are a useful tool in lattice gauge theory and we illustrate this by showing that our results can describe data from two color quantum chromodynamics simulations with chemical potential in the symplectic class

a bound for global solutions of semilinear heat equations we show that global positive solutions of the initial boundary value problem foru t u u p are bounded provided thatp 1 is subcritical our bound depends only on sup norm of the initial data and is useful to classify initial data by the asymptotic behavior of the solutions as time tend to infinity

the distribution of lyapunov exponents exact results for random matrices simple exact expressions are derived for all the lyapunov exponents of certainn dimensional stochastic linear dynamical systems in the case of the product of independent random matrices each of which has independent gaussian entries with mean zero and variance 1 n the exponents have an exponential distribution asn in the case of the time ordered product integral of exp n 1 2dw where the entries of then n matrixw t are independent standard wiener processes the exponents are equally spaced for fixedn and thus have a uniform distribution as n

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

lyapunov exponents for some isotropic random matrix ensembles a random matrix with rows distributed as a function of their length is said to be isotropic when these distributions are gaussian beta type i or beta type ii previous work has from the viewpoint of integral geometry obtained the explicit form of the distribution of the determinant we use these result to evaluate the sum of the lyapunov spectrum of the corresponding random matrix product and we further give explicit expressions for the largest lyapunov exponent generalisations to the case of complex or quaternion entries are also given for standard gaussian matrices x the full lyapunov spectrum for products of random matrices i n 1 over c x is computed in terms of a generalised hypergeometric function in general and in terms of a single single integral involving a modified bessel function for the largest lyapunov exponent

blowup solutions for a reaction diffusion system with exponential nonlinearities abstract we consider the following parabolic system whose nonlinearity has no gradient structure t u u e p v t v v e q u u 0 u 0 v 0 v 0 p q 0 in the whole space r n we show the existence of a stable blowup solution and obtain a complete description of its singularity formation the construction relies on the reduction of the problem to a finite dimensional one and a topological argument based on the index theory to conclude in particular our analysis uses neither the maximum principle nor the classical methods based on energy type estimates which are not supported in this system the stability is a consequence of the existence proof through a geometrical interpretation of the quantities of blowup parameters whose dimension is equal to the dimension of the finite dimensional problem

on the largest lyapunov exponent for products of gaussian matrices the paper provides a new integral formula for the largest lyapunov exponent of gaussian matrices which is valid in the real complex and quaternion valued cases this formula is applied to derive asymptotic expressions for the largest lyapunov exponent when the size of the matrix is large and compare the lyapunov exponents in models with a spike and no spikes

isotropic brownian motions over complex fields as a solvable model for may wigner stability analysis we consider matrix valued stochastic processes known as isotropic brownian motions and show that these can be solved exactly over complex fields while these processes appear in a variety of questions in mathematical physics our main motivation is their relation to a may wigner like stability analysis for which we obtain a stability phase diagram the exact results establish the full joint probability distribution of the finite time lyapunov exponents and may be used as a starting point for a more detailed analysis of the stability instability phase transition our derivations rest on an explicit formulation of a fokker planck equation for the lyapunov exponents this formulation happens to coincide with an exactly solvable class of models of the calgero sutherland type originally encountered for a model of phase coherent transport the exact solution over complex fields describes a determinantal point process of biorthogonal type similar to recent results for products of random matrices and is also closely related to hermitian matrix models with an external source

hidden markov processes an overview of statistical and information theoretic aspects of hidden markov processes hmps is presented an hmp is a discrete time finite state homogeneous markov chain observed through a discrete time memoryless invariant channel in recent years the work of baum and petrie 1966 on finite state finite alphabet hmps was expanded to hmps with finite as well as continuous state spaces and a general alphabet in particular statistical properties and ergodic theorems for relative entropy densities of hmps were developed consistency and asymptotic normality of the maximum likelihood ml parameter estimator were proved under some mild conditions similar results were established for switching autoregressive processes these processes generalize hmps new algorithms were developed for estimating the state parameter and order of an hmp for universal coding and classification of hmps and for universal decoding of hidden markov channels these and other related topics are reviewed

blow up profile for the complex ginzburg landau equation abstract we construct a solution to the complex ginzburg landau equation which blows up in finite time t only at one blow up point we also give a sharp description of its blow up profile the proof relies on the reduction of the problem to a finite dimensional one and the use of index theory to conclude two major difficulties arise in the proof the linearized operator around the profile is not self adjoint and it has a second neutral mode in the last section the interpretation of the parameters of the finite dimensional problem in terms of the blow up time and the blow up point gives the stability of the constructed solution with respect to perturbations in the initial data

lyapunov exponents of free operators abstract lyapunov exponents of a dynamical system are a useful tool to gauge the stability and complexity of the system this paper offers a definition of lyapunov exponents for a sequence of free linear operators the definition is based on the concept of the extended fuglede kadison determinant we establish the existence of lyapunov exponents derive formulas for their calculation and show that lyapunov exponents of free variables are additive with respect to operator product we illustrate these results using an example of free operators whose singular values are distributed by the marchenko pastur law and relate this example to c m newman s triangle law for the distribution of lyapunov exponents of large random matrices with independent gaussian entries as an interesting by product of our results we derive a relation between the extended fuglede kadison determinant and voiculescu s s transform

calculation of some determinants using the s shifted factorial several determinants with gamma functions as elements are evaluated these kinds of determinants are encountered for example in the computation of the probability density of the determinant of random matrices the s shifted factorial is defined as a generalization for non negative integers of the power function the rising factorial or pochammer s symbol and the falling factorial it is a special case of a polynomial sequence of the binomial type studied in combinatorics theory in terms of the gamma function an extension is defined for negative integers and even complex values properties mainly composition laws and binomial formulae are given they are used to evaluate families of generalized vandermonde determinants with s shifted factorials as elements instead of power functions

exact solutions for ising chains in a random field the authors present an exact solution for the one dimensional ising model in a random field the distribution of field strengths is site independent symmetric and has a three peak structure the free energy is obtained for all finite temperatures the low temperature behaviour is studied in detail they find an expansion for the free energy in integer powers of temperature the ground state energy and the zero point entropy are calculated explicitly the specific heat is linear for low temperatures in contradiction with mean field theory the origin of an additional exponential parameter is discussed

universality of local spectral statistics of products of random matrices we derive exact analytical expressions for correlation functions of singular values of the product of m ginibre matrices of size n in the double scaling limit m n ensuremath rightarrow ensuremath infty the singular value statistics is described by a determinantal point process with a kernel that interpolates between gaussian unitary ensemble statistic and dirac delta picket fence statistic in the thermodynamic limit n ensuremath rightarrow ensuremath infty the interpolation parameter is given by the limiting quotient a n m one of our goals is to find an explicit form of the kernel at the hard edge in the bulk and at the soft edge for any a we find that in addition to the standard scaling regimes there is a transitional regime which interpolates between the hard edge and the bulk we conjecture that these results are universal and that they apply to a broad class of products of random matrices from the gaussian basin of attraction including correlated matrices we corroborate this conjecture by numerical simulations additionally we show that the local spectral statistics of the considered random matrix products is identical with the local statistics of dyson brownian motion with the initial condition given by equidistant positions with the crucial difference that this equivalence holds only locally finally we have identified a mesoscopic spectral scale at the soft edge which is crucial for the unfolding of the spectrum

the quantum group structure associated with non linearly extended virasoro algebras recently an infinite family of chiral virasoro vertex operators whose exchange algebra is given by the universalr matrix ofsl 2 q has been constructed in the present paper the case of non linearly w extended virasoro symmetries related to the algebrasan n 1 is considered along the same line contrary to the previous case a1 the standardr matrix ofsl n 1 q does not come out and a different solution of yang and baxter s equations is derived the associated quantum group structure is displayed

determining lyapunov exponents from a time series we present the first algorithms that allow the estimation of non negative lyapunov exponents from an experimental time series lyapunov exponents which provide a qualitative and quantitative characterization of dynamical behavior are related to the exponentially fast divergence or convergence of nearby orbits in phase space a system with one or more positive lyapunov exponents is defined to be chaotic our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems we monitor the long term growth rate of small volume elements in an attractor the method is tested on model systems with known lyapunov spectra and applied to data for the belousov zhabotinskii reaction and couette taylor flow

exact solutions for spectra and green s functions in random one dimensional systems for chains of harmonic oscillators with random masses a set of equations is derived which determine the spatial fourier components of the average one particle green s function these equations are valid for complex values of the frequency a relation between the spectral density and functions introduced by schmidt is discussed exact solutions for this green s function and the less complicated characteristics function the analytic continuation into the complex frequency plane of the accumulated spectral density and the inverse localization length of the eigenfunctions are derived for exponential distributions of the masses for some cases the characteristic function is calculated numerically for gamma distributions the equations are cast in the form of ordinary higher order differential equations these have been solved numerically for determining the characteristic function for arbitrary mass distributions a cumulant expansion and a peculiar symmetry of the green s function are discussed the method is also applied to chains where the spring constants and or the masses have random values also for these systems exact solutions are discussed for exponential distributions e g of both masses and spring constants the characteristic function is expressed in bessel functions the relation with certain random relaxation models is shown finally x y hamiltonians with random exchange constants and or magnetic fields or equivalently tight binding electron models with diagonal and or off diagonal disorder are considered here the green s function does not depend on the wave number if the distribution of exchange constants is symmetric around the origin new solutions for the characteristic function and green s function are derived for a number of cases including exponentially distributed magnetic fields and power law distributed exchange constants

convergence asymptotic periodicity and finite point blow up in one dimensional semilinear heat equations abstract we consider the initial value problem for the semilinear heat equation u t u xx f u t 0 x l t 0 under the dirichlet the neumann or the periodic boundary conditions we show that each solution whether it exists globally for t 0 or blows up in a finite time possesses an asymptotic profile in a certain sense and tends to this profile as time increases in the special case where f u t t f u t for some t 0 among other things the above statement is interpreted as saying that any bounded global solution converges as t to a time t periodic solution having some specific spatial structure in the case where the solution blows up in a finite time say at t t 0 assuming simply that f is a smooth function satisfying some growth conditions and that the initial data is a nonconstant bounded function we prove that the blow up set is a finite set and that lim t t 0 u x t x exists with being a smooth function having at most finitely many singular points

commutative law for products of infinitely large isotropic random matrices ensembles of isotropic random matrices are defined by the invariance of the probability measure under the left and right multiplication by an arbitrary unitary matrix we show that the multiplication of large isotropic random matrices is spectrally commutative and self averaging in the limit of infinite matrix size n the notion of spectral commutativity means that the eigenvalue density of a product abc of such matrices is independent of the order of matrix multiplication for example the matrix abcd has the same eigenvalue density as adcb in turn the notion of self averaging means that the product of n independent but identically distributed random matrices which we symbolically denote by aaa has the same eigenvalue density as the corresponding power a n of a single matrix drawn from the underlying matrix ensemble for example the eigenvalue density of abccabc is the same as that of a 2 b 2 c 3 we also discuss the singular behavior of the eigenvalue and singular value densities of isotropic matrices and their products for small eigenvalues 0 we show that the singularities at the origin of the eigenvalue density and of the singular value density are in one to one correspondence in the limit n the eigenvalue density of an isotropic random matrix has a power law singularity at the origin s with a power se 0 2 when and only when the density of its singular values has a power law singularity with a power s 4 s these results are obtained analytically in the limit n we supplement these results with numerical simulations for large but finite n and discuss finite size effects for the most common ensembles of isotropic random matrices

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

the zero set of a solution of a parabolic equation on etudie l ensemble nul d une solution u t x de l equation u t a x t u xx b x t u x c x t u sous des hypotheses tres generales sur les coefficients a b et c

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

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

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

maximal lyapunov exponents for random matrix products in this article we study the lyapunov exponent for random matrix products of positive matrices and express them in terms of associated complex functions this leads to new explicit formulae for the lyapunov exponents and to an efficient method for their computation

universality conjecture and results for a model of several coupled positive definite matrices the paper contains two main parts in the first part we analyze the general case of p geq 2 matrices coupled in a chain subject to cauchy interaction similarly to the itzykson zuber interaction model the eigenvalues of the cauchy chain form a multi level determinantal point process we first compute all correlations functions in terms of cauchy biorthogonal polynomials and locate them as specific entries of a p 1 times p 1 matrix valued solution of a riemann hilbert problem in the second part we fix the external potentials as classical laguerre weights we then derive strong asymptotics for the cauchy biorthogonal polynomials when the support of the equilibrium measures contains the origin as a result we obtain a new family of universality classes for multi level random determinantal point fields which include the bessel nu universality for 1 level and the meijer g universality for 2 level our analysis uses the deift zhou nonlinear steepest descent method and the explicit construction of a p 1 times p 1 origin parametrix in terms of meijer g functions the solution of the full riemann hilbert problem is derived rigorously only for p 3 but the general framework of the proof can be extended to the cauchy chain of arbitrary length p

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

construction of a stable periodic solution to a semilinear heat equation with a prescribed profile we construct a periodic solution to the semilinear heat equation with power nonlinearity in one space dimension which blows up in finite time t only at one blow up point we also give a sharp description of its blow up profile the proof relies on the reduction of the problem to a finite dimensional one and the use of index theory to conclude thanks to the geometrical interpretation of the finite dimensional parameters in terms of the blow up time and blow up point we derive the stability of the constructed solution with respect to initial data

solution of a nonlinear heat equation with arbitrarily given blow up points we consider the equation where i u is scalar valued and p 1 it has been proven that if u t blows up at time t the blow up points are finite in number and located in i our aim is to prove that this result is optimal that is for any given points x1 xk in i there is a solution u such that its blow up points are exactly x1 xk

critical exponent for the bipolar blowup in a semilinear parabolic equation abstract the blowup of solutions of the initial boundary value problem formula is studied where is a cone inrn we say that a solutionuexhibits bipolar blowup if formula for somet 1 for the bipolar blowup in the following sense if 1 p then the bipolar blowup does not occur for any sufficiently small initial data the value ofp is expressed in terms of the dimensionnand the second dirichlet eigenvalue of the laplace beltrami operator on sn 1 in the case of r we definek polar blowup and determine critical exponents for thek polar blowup fork 1 2 3

single point blow up for a semilinear initial value problem abstract the initial value problem on r r is considered u t t x u xx t x u t x u t r 0 u 0 x x where 0 and is a fixed large number it is known that for some initial values the solution u t x exists only up to some finite time t and that u t as t t for the specific initial value k where 0 xx 0 r 0 k is sufficiently large it is shown that if x 0 then lim t t u t x and lim t t u x t x exist and are finite in other words blow up occurs only at the point x 0

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