tridiagonal realization of the antisymmetric gaussian β ensemble the householder reduction of a member of the antisymmetric gaussian unitary ensemble gives an antisymmetric tridiagonal matrix with all independent elements the random variables permit the introduction of a positive parameter and the eigenvalue probability density function of the corresponding random matrices can be computed explicitly as can the distribution of qi the first components of the eigenvectors three proofs are given one involves an inductive construction based on bordering of a family of random matrices which are shown to have the same distributions as the antisymmetric tridiagonal matrices this proof uses the dixon anderson integral from selberg integral theory a second proof involves the explicit computation of the jacobian for the change of variables between real antisymmetric tridiagonal matrices its eigenvalues and qi the third proof maps matrices from the antisymmetric gaussian ensemble to those realizing particular examples of the laguerre ensemble in addition to t

exact operator solution of the calogero sutherland model the wave functions of the calogero sutherland model are known to be expressible in terms of jack polynomials a formula which allows to obtain the wave functions of the excited states by acting with a string of creation operators on the wave function of the ground state is presented and derived the creation operators that enter in this formula of rodrigues type for the jack polynomials involve dunkl operators

applications and generalizations of fisher hartwig asymptotics fisher hartwig asymptotics refers to the large n form of a class of toeplitz determinants with singular generating functions this class of toeplitz determinants occurs in the study of the spin spin correlations for the two dimensional ising model and the ground state density matrix of the impenetrable bose gas amongst other problems in mathematical physics we give a new application of the original fisher hartwig formula to the asymptotic decay of the ising correlations above tc while the study of the bose gas density matrix leads us to generalize the fisher hartwig formula to the asymptotic form of random matrix averages over the classical groups and the gaussian and laguerre unitary matrix ensembles another viewpoint of our generalizations is that they extend to hankel determinants the fisher hartwig asymptotic form known for toeplitz determinants

eigenvalue separation in some random matrix models the eigenvalue density for members of the gaussian orthogonal and unitary ensembles follows the wigner semicircle law if the gaussian entries are all shifted by a constant amount s 2n 1 2 where n is the size of the matrix in the large n limit a single eigenvalue will separate from the support of the wigner semicircle provided s 1 in this study using an asymptotic analysis of the secular equation for the eigenvalue condition we compare this effect to analogous effects occurring in general variance wishart matrices and matrices from the shifted mean chiral ensemble we undertake an analogous comparative study of eigenvalue separation properties when the sizes of the matrices are fixed and s and higher rank analogs of this setting this is done using exact expressions for eigenvalue probability densities in terms of generalized hypergeometric functions and using the interpretation of the latter as a green function in the dyson brownian motion model for the shifted mean gaussian unitary ensemble an

on the sutherland spin model of b n type and its associated spin chain the b n hyperbolic sutherland spin model is expressed in terms of a suitable set of commuting dunkl operators this fact is exploited to derive a complete family of commuting integrals of motion of the model thus establishing its integrability the dunkl operators are shown to possess a common flag of invariant finite dimensional linear spaces of smooth scalar functions this implies that the hamiltonian of the model preserves a corresponding flag of smooth spin functions the discrete spectrum of the restriction of the hamiltonian to this spin flag is explicitly computed by triangularization the integrability of the hyperbolic sutherland spin chain of b n type associated with the dynamical model is proved using polychronakos s freezing trick

certain hypergeometric series related to the root system bc we show that the generalized hypergeometric function 2 f 1 of matrix argument is the series expansion at the origin of a special case of the hypergeometric function associated with the root system of type bc in addition we prove that the jacobi polynomials of matrix argument correspond to the jacobi polynomials associated with the root system of type bc we also give a precise relation between jack polynomials and the jacobi polynomials associated with the root system of type a as a side result one obtains generalized hook length formulas which are related to harish chandra s c function and one can prove a conjecture due to macdonald relating two inner products on a space of symmetric functions

riesz distributions and laplace transform in the dunkl setting of type a abstract we study riesz distributions in the framework of rational dunkl theory associated with root systems of type a as an important tool we employ a laplace transform involving the associated dunkl kernel which essentially goes back to macdonald 18 but was so far only established at a formal level we give a rigorous treatment of this transform based on suitable estimates of the type a dunkl kernel our main result is a precise analogue in the dunkl setting of a well known result by gindikin stating that a riesz distribution on a symmetric cone is a positive measure if and only if its exponent is contained in the wallach set for riesz distributions in the dunkl setting we obtain an analogous characterization in terms of a generalized wallach set which depends on the multiplicity parameter on the root system

some combinatorial properties of jack symmetric functions if 1 l n then write 2 n or 1 n if p is another partition then write p c j if pi 6 for all i i e if the diagram of 1 contains the diagram of p if ipl ii 1 then write p 2 1 reverse lexicographic order if either j i or the first nonvanishing difference e pi is positive for instance writing 2 1 i for ai i it 5 5 41 32 5 311 221 2111 11111 finally write p i if ipl lal and p1 pl2 pli a1 a2 li for all i macdonald m p 63 call the partial ordering the natural ordering but we will call it the dominance ordering let x x x2 be an infinite set of indeterminates as in mi we

duality of orthogonal and symplectic matrix integrals and quaternionic feynman graphs we present an asymptotic expansion for quaternionic self adjoint matrix integrals the feynman diagrams appearing in the expansion are ordinary ribbon graphs and their non orientable counterparts we show that the 2n 2n gaussian orthogonal ensemble goe and n n gaussian symplectic ensemble gse have exactly the same expansion term by term except that the contributions from graphs on a non orientable surface with odd euler characteristic carry the opposite sign as an application we give a new topological proof of the known duality for correlations of characteristic polynomials demonstrating that this duality is equivalent to poincare duality of graphs drawn on a compact surface another consequence of our graphical expansion formula is a simple and simultaneous re derivation of the central limit theorem for goe gue gaussian unitary ensemble and gse the three cases have exactly the same graphical limiting formula except for an overall constant that represents the type of the ensemble

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

explicit formulae for the generalized hermite polynomials in superspace we provide explicit formulae for the orthogonal eigenfunctions of the supersymmetric extension of the rational calogero moser sutherland model with harmonic confinement i e the generalized hermite or hi jack polynomials in superspace the construction relies on the triangular action of the hamiltonian on the supermonomial basis this translates into determinantal expressions for the hamiltonian s eigenfunctions

on random matrices from the compact classical groups if m is a matrix taken randomly with respect to normalized haar measure on u n o n or sp n then the real and imaginary parts of the random variables tr mk k 1 converge to independent normal random variables with mean zero and variance k 2 as the size n of the matrix goes to infinity for the unitary group this is a direct consequence of the strong szeg6 theorem for toeplitz determinants we will prove a conjecture of diaconis

quantum versus classical integrability in calogero moser systems calogero moser systems are classical and quantum integrable multiparticle dynamics defined for any root system the quantum calogero systems having 1 q2 potential and a confining q2 potential and the sutherland systems with 1 sin2q potentials have integer energy spectra characterized by the root system various quantities of the corresponding classical systems e g minimum energy frequencies of small oscillations the eigenvalues of the classical lax pair matrices etc at the equilibrium point of the potential are investigated analytically as well as numerically for all root systems to our surprise most of these classical data are also integers or they appear to be quantized to be more precise these quantities are polynomials of the coupling constant s with integer coefficients the close relationship between quantum and classical integrability in calogero moser systems deserves fuller analytical treatment which would lead to better understanding of these systems and of integrable systems in general

the averaged characteristic polynomial for the gaussian and chiral gaussian ensembles with a source in classical random matrix theory the gaussian and chiral gaussian random matrix models with a source are realized as shifted mean gaussian and chiral gaussian random matrices with real 1 complex 2 and real quaternion 4 elements we use the dyson brownian motion model to give a meaning for general 0 in the gaussian case a further construction valid for 0 is given as the eigenvalue pdf of a recursively defined random matrix ensemble in the case of real or complex elements a combinatorial argument is used to compute the averaged characteristic polynomial the resulting functional forms are shown to be special cases of duality formulas due to desrosiers new derivations of the general case of desrosiers dualities are given a soft edge scaling limit of the averaged characteristic polynomial is identified and an explicit evaluation in terms of so called incomplete airy functions is obtained

eigenvalues of ruijsenaars schneider models associated with a n 1 root system in bethe ansatz formalism ruijsenaars schneider models associated with an 1 root system with a discrete coupling constant are studied the eigenvalues of the hamiltonian are given in terms of the bethe ansatz formulas taking the nonrelativistic limit we obtain the spectrum of the corresponding calogero moser systems in the third formulas of felder et al

lp fourier multipliers for the dunkl operator on the real line we consider fourier multipliers for lp associated with the dunkl operator on r and establish a version of hormander s multiplier theorem in applying this version we come up with some results regarding the oscillating multipliers partial sum operators and generalized bessel potentials

on invariant 2 2 β ensembles of random matrices abstract we introduce and solve exactly a family of invariant 2 2 random matrices depending on one parameter and we show that rotational invariance and real dyson index are not incompatible properties the probability density for the entries contains a weight function and a multiple trace trace interaction term which corresponds to the representation of the vandermonde squared coupling on the basis of power sums as a result the effective dyson index eff of the ensemble can take any real value in an interval two weight functions gaussian and non gaussian are explored in detail and the connections with ensembles of dumitriu edelman and the so called poisson wigner crossover for the level spacing are respectively highlighted a curious spectral twinning between ensembles of different symmetry classes is unveiled as a consequence the identification between symmetry group orthogonal unitary or symplectic and the exponent of the vandermonde 1 2 4 is shown to be potentially deceptive the proposed technical tool more generically allows for designing actual matrix models which i are rotationally invariant ii have a real dyson index eff iii have a pre assigned confining potential or alternatively level spacing profile the analytical results have been checked through numerical simulations with an excellent agreement eventually we discuss possible generalizations and further directions of research

random matrix ensembles with an effective extensive external charge recent theoretical studies of chaotic scattering have encountered ensembles of random matrices in which the eigenvalue probability density function contains a one body factor with an exponent proportional to the number of eigenvalues two such ensembles have been encountered an ensemble of unitary matrices specified by the so called poisson kernel and the laguerre ensemble of positive definite matrices here we consider various properties of these ensembles jack polynomial theory is used to prove a reproducing property of the poisson kernel and a certain unimodular mapping is used to demonstrate that the variance of a linear statistic is the same as in the dyson circular ensemble for the laguerre ensemble the scaled global density is calculated exactly for all even values of the parameter while for random matrices with unitary symmetry the neighbourhood of the smallest eigenvalue is shown to be in the soft edge universality class

global spectrum fluctuations for the β hermite and β laguerre ensembles via matrix models we study the global spectrum fluctuations for hermite and laguerre ensembles via the tridiagonal matrix models introduced previously by the present authors j math phys 43 5830 2002 and prove that the fluctuations describe a gaussian process on polynomials we extend our results to slightly larger classes of random matrices

coupled principal component analysis a framework for a class of coupled principal component learning rules is presented in coupled rules eigenvectors and eigenvalues of a covariance matrix are simultaneously estimated in coupled equations coupled rules can mitigate the stability speed problem affecting noncoupled learning rules since the convergence speed in all eigendirections of the jacobian becomes widely independent of the eigenvalues of the covariance matrix a number of coupled learning rule systems for principal component analysis two of them new is derived by applying newton s method to an information criterion the relations to other systems of this class the adaptive learning algorithm ala the robust recursive least squares algorithm rrlsa and a rule with explicit renormalization of the weight vector length are established

laguerre functions on symmetric cones and recursion relations in the real case in this article we derive differential recursion relations for the laguerre functions on the cone of positive definite real matrices the highest weight representations of the group sp n r play a fundamental role each such representation acts on a hilbert space of holomorphic functions on the tube domain isym n r we then use the laplace transform to carry the lie algebra action over to l2 d v the differential recursion relations result by restricting to a distinguished three dimensional subalgebra which is isomorphic to s 2 r

interacting particles on the line and dunkl intertwining operator of type a application to the freezing regime we consider a one dimensional system of brownian particles that repel each other through a logarithmic potential we study two formulations for the system and the relation between them the first dyson s brownian motion model has an interaction coupling constant determined by the parameter 0 when 1 2 and 4 this model can be regarded as a stochastic realization of the eigenvalue statistics of gaussian random matrices the second system comes from dunkl processes which are defined using differential difference operators dunkl operators associated with finite abstract vector sets called root systems when the type a root system is specified dunkl processes constitute a one parameter system similar to dyson s model with the difference that its particles interchange positions spontaneously we prove that the type a dunkl processes with parameter k 0 starting from any symmetric initial configuration are equivalent to dyson s model with the parameter 2k we focus on the intertwining operators since they play a central role in the mathematical theory of dunkl operators but their general closed form is not yet known using the equivalence between symmetric dunkl processes and dyson s model we extract the effect of the intertwining operator of type a on symmetric polynomials from these processes transition probability densities in the strong coupling limit the intertwining operator maps all symmetric polynomials onto a function of the sum of their variables in this limit dyson s model freezes and it becomes a deterministic process with a final configuration proportional to the roots of the hermite polynomials multiplied by the square root of the process time while being independent of the initial configuration

matrix models for beta ensembles this paper constructs tridiagonal random matrix models for general 0 hermite gaussian and laguerre wishart ensembles these generalize the well known gaussian and wishart models for 1 2 4 furthermore in the cases of the laguerre ensembles we eliminate the exponent quantization present in the previously known models we further discuss applications for the new matrix models and present some open problems

the calogero sutherland model and generalized classical polynomials multivariable generalizations of the classical hermite laguerre and jacobi polynomials occur as the polynomial part of the eigenfunctions of certain schrodinger operators for calogero sutherland type quantum systems for the generalized hermite and laguerre polynomials the multidimensional analogues of many classical results regarding generating functions differentiation and integration formulas recurrence relations and summation theorems are obtained we use this and related theory to evaluate the global limit of the ground state density obtaining in the hermite case the wigner semi circle law and to give an explicit solution for an initial value problem in the hermite and laguerre case

quantum calogero moser models integrability for all root systems the issues related to the integrability of quantum calogero moser models based on any root systems are addressed for the models with degenerate potentials i e the rational with without the harmonic confining force the hyperbolic and the trigonometric we demonstrate the following for all the root systems i construction of a complete set of quantum conserved quantities in terms of a total sum of the lax matrix l i e sigma mu nu is an element ofr l n mu nu in which r is a set of r r vectors invariant under the action of the coxeter group they form a single coxeter orbit ii proof of liouville integrability iii triangularity of the quantum hamiltonian and the entire discrete spectrum generalized jack polynomials are defined for all root systems as unique eigenfunctions of the hamiltonian iv equivalence of the lax operator and the dunkl operator v algebraic construction of all excited states in terms of creation operators these are mainly generalizations of the results known for the models based on the a series i e su n type root systems

the yangian symmetry in the spin calogero model and its applications by using the non symmetric hermite polynomials and a technique based on the yangian gelfand zetlin bases we decompose the space of states of the calogero model with spin into irreducible yangian modules construct an orthogonal basis of eigenvectors and derive product type formulae for norms of these eigenvectors

exchange operator formalism for n body spin models with near neighbours interactions we present a detailed analysis of the spin models with near neighbours interactions constructed in our previous paper enciso et al 2005 phys lett b 605 214 by a suitable generalization of the exchange operator formalism we provide a complete description of a certain flag of finite dimensional spaces of spin functions preserved by the hamiltonian of each model by explicitly diagonalizing the hamiltonian in the latter spaces we compute several infinite families of eigenfunctions of the above models in closed form in terms of generalized laguerre and jacobi polynomials

on the statistical mechanics approach in the random matrix theory integrated density of states we consider the ensemble of random symmetricn n matrices specified by an orthogonal invariant probability distribution we treat this distribution as a gibbs measure of a mean field type model this allows us to show that the normalized eigenvalue counting function of this ensemble converges in probability to a nonrandom limit asn and that this limiting distribution is the solution of a certain self consistent equation

critical statistics in quantum chaos and calogero sutherland model at finite temperature we investigate the spectral properties of a generalized gaussian orthogonal ensemble capable of describing critical statistics the joint distribution of eigenvalues of this model is expressed as the diagonal element of the density matrix of a gas of particles governed by the calogero sutherland cs hamiltonian taking advantage of the correspondence between cs particles and eigenvalues and utilizing a recently conjectured expression by kravtsov and tsvelik for the finite temperature density density correlations of the cs model we show that the number variance of our random matrix model is asymptotically linear with a slope depending on the parameters of the model such linear behavior is a signature of critical statistics this random matrix model may be relevant for the description of spectral correlations of complex quantum systems with a self similar or fractal poincare section of its classical counterpart this is shown in detail for two examples the anisotropic kepler problem and a kicked particle in a well potential in both cases the number variance and the delta 3 statistic are accurately described by our analytical results

complex singular wishart matrices and applications in this paper complex singular wishart matrices and their applications to information theory are investigated in particular a volume element on the space of positive semidefinite m x m hermitian matrices of rank n m is introduced and some transformation properties are established the jacobian for the change of variables in the singular value decomposition of general m x n complex matrices is derived then the density functions are formulated for all rank n complex singular wishart distributions from this the joint eigenvalue density of low rank complex wishart matrices are derived the derived densities are used to evaluate the most important information theoretic measure the so called ergodic channel capacity of multiple input multiple output mimo spatially correlated rayleigh distributed wireless communication channels

two step asymptotics of scaled dunkl processes dunkl processes are generalizations of brownian motion obtained by using the differential difference operators known as dunkl operators as a replacement of spatial partial derivatives in the heat equation special cases of these processes include dyson s brownian motion model and the wishart laguerre eigenvalue processes which are well known in random matrix theory it is known that the dynamics of dunkl processes is obtained by transforming the heat kernel using dunkl s intertwining operator it is also known that under an appropriate scaling their distribution function converges to a steady state distribution which depends only on the coupling parameter as the process time t tends to infinity we study scaled dunkl processes starting from an arbitrary initial distribution and we derive expressions for the intertwining operator in order to calculate the asymptotics of the distribution function in two limiting situations in the first one is fixed and t tends to infinity approach to the steady s

generalized hermite polynomials and the heat equation for dunkl operators based on the theory of dunkl operators this paper presents a general concept of multivariable hermite polynomials and hermite functions which are associated with finite reflection groups on n the definition and properties of these generalized hermite systems extend naturally those of their classical counterparts partial derivatives and the usual exponential kernel are here replaced by dunkl operators and the generalized exponential kernel k of the dunkl transform in the case of the symmetric group s n our setting includes the polynomial eigenfunctions of certain calogero sutherland type operators the second part of this paper is devoted to the heat equation associated with dunkl s laplacian as in the classical case the corresponding cauchy problem is governed by a positive one parameter semigroup this is assured by a maximum principle for the generalized laplacian the explicit solution to the cauchy problem involves again the kernel k which is on the way proven to be nonnegative for real arguments

littlewood paley operators associated with the dunkl operator on r we consider a vector version of l p multipliers for the dunkl transform on r and we prove l p inequalities for the littlewood paley operators

explicit formulae for the eigenfunctions of the n body calogero model we consider the quantum calogero model which describes n non distinguishable quantum particles on the real line confined by a harmonic oscillator potential and interacting via two body interactions proportional to the inverse square of the inter particle distance we elaborate a novel solution algorithm which allows us to obtain fully explicit formulae for its eigenfunctions arbitrary coupling parameter and particle number we also show that our method applies with minor changes to all calogero models associated with classical root systems

common algebraic structure for the calogero sutherland models we investigate a common algebraic structure for the rational and trigonometric calogero sutherland models by using the exchange operator formalism we show that the set of jack polynomials whose arguments are dunkl type operators provides an orthogonal basis for the rational case

sample sizes for constructing confidence intervals and testing hypotheses although estimation and confidence intervals have become popular alternatives to hypothesis testing and p values statisticians usually determine sample sizes for randomized clinical trials by controlling the power of a statistical test at an appropriate alternative even those statisticians who recommend the use of confidence intervals for inference there is merit in achieving consistency in the techniques for data analysis and sample size determination to that end this paper compares sample size determination with use of the length of the confidence interval with that obtained by control of power

exact results and universal asymptotics in the laguerre random matrix ensemble for various values of the parameters in the laguerre random matrix ensemble the distribution of the smallest eigenvalue and the scaled n level distribution function are calculated exactly in terms of generalized hypergeometric functions in certain cases these functions are expressed as multidimensional integrals from which asymptotic formulas are calculated and predictions of universal behavior verified

a random matrix decimation procedure relating β 2 r 1 to β 2 r 1 classical random matrix ensembles with orthogonal symmetry have the property that the joint distribution of every second eigenvalue is equal to that of a classical random matrix ensemble with symplectic symmetry these results are shown to be the case r 1 of a family of inter relations between eigenvalue probability density functions for generalizations of the classical random matrix ensembles referred to as ensembles the inter relations give that the joint distribution of every r 1 st eigenvalue in certain ensembles with 2 r 1 is equal to that of another ensemble with 2 r 1 the proof requires generalizing a conditional probability density function due to dixon and anderson

the representation ring of the unitary groups and markov processes of algebraic origin abstract the paper consists of two parts the first part introduces the representation ring for the family of compact unitary groups u 1 u 2 this novel object is a commutative graded algebra r with infinite dimensional homogeneous components it plays the role of the algebra of symmetric functions which serves as the representation ring for the family of finite symmetric groups the purpose of the first part is to elaborate on the basic definitions and prepare the ground for the construction of the second part of the paper the second part deals with a family of markov processes on the dual object to the infinite dimensional unitary group u these processes were defined in a joint work with alexei borodin 2012 5 the main result of the present paper consists in the derivation of an explicit expression for their infinitesimal generators it is shown that the generators are implemented by certain second order partial differential operators with countably many variables initially defined as operators on r

sparse regular random graphs spectral density and eigenvectors we examine the empirical distribution of the eigenvalues and the eigenvectors of adjacency matrices of sparse regular random graphs we find that when the degree sequence of the graph slowly increases to infinity with the number of vertices the empirical spectral distribution converges to the semicircle law moreover we prove concentration estimates on the number of eigenvalues over progressively smaller intervals we also show that with high probability all the eigenvectors are delocalized