raney distributions and random matrix theory recent works have shown that the family of probability distributions with moments given by the fuss catalan numbers permit a simple parameterized form for their density we extend this result to the raney distribution which by definition has its moments given by a generalization of the fuss catalan numbers such computations begin with an algebraic equation satisfied by the stieltjes transform which we show can be derived from the linear differential equation satisfied by the characteristic polynomial of random matrix realizations of the raney distribution for the fuss catalan distribution an equilibrium problem characterizing the density is identified the stieltjes transform for the limiting spectral density of the singular values squared of the matrix product formed from q inverse standard gaussian matrices and s standard gaussian matrices is shown to satisfy a variant of the algebraic equation relating to the raney distribution supported on 0 infty we show that it too permits a simple functional form upon the introduction of an appropriate choice of parameterization as an application the leading asymptotic form of the density as the endpoints of the support are approached is computed and is shown to have some universal features

uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory we consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn x dx e nv x dx on the line as n the potentials v are assumed to be real analytic with sufficient growth at infinity the principle results concern plancherel rotach type asymptotics for the orthogonal polynomials down to the axis using these asymptotics we then prove universality for a variety of statistical quantities arising in the theory of random matrix models some of which have been considered recently in 31 and also in 4 an additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials see also 4 the orthogonal polynomial problem is formulated as a riemann hilbert problem following 19 20 the riemann hilbert problem is analyzed in turn using the steepest descent method introduced in 12 and further developed in 11 13 a critical role in our method is played by the equilibrium measure d v for v as analyzed in 8 1999 john wiley sons inc

dyson s nonintersecting brownian motions with a few outliers consider n nonintersecting brownian particles on r dyson brownian motions all starting from the origin at time t 0 and forced to return to x 0 at time t 1 for large n the average mean density of particles has its support for each 0 0 does it affect the brownian fluctuations along the curve c for large n in this paper we show that no new process appears as long as one considers points y t is an element of c such that 0 infinity this kernel is closely related to one found by baik ben arous and peche in the context of multivariate statistics c 2008 wiley periodicals inc

lifting valid inequalities for the precedence constrained knapsack problem this paper considers the precedence constrained knapsack problem more specifically we are interested in classes of valid inequalities which are facet defining for the precedence constrained knapsack polytope we study the complexity of obtaining these facets using the standard sequential lifting procedure applying this procedure requires solving a combinatorial problem for valid inequalities arising from minimal induced covers we identify a class of lifting coefficients for which this problem can be solved in polynomial time by using a supermodular function and for which the values of the lifting coefficients have a combinatorial interpretation for the remaining lifting coefficients it is shown that this optimization problem is strongly np hard the same lifting procedure can be applied to 1 k configurations although in this case the same combinatorial interpretation no longer applies we also consider k covers to which the same procedure need not apply in general we show that facets of the polytope can still be generated using a similar lifting technique for tree knapsack problems we observe that all lifting coefficients can be obtained in polynomial time computational experiments indicate that these facets significantly strengthen the lp relaxation

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

statistical properties of random density matrices statistical properties of ensembles of random density matrices are investigated we compute traces and von neumann entropies averaged over ensembles of random density matrices distributed according to the bures measure the eigenvalues of the random density matrices are analysed we derive the eigenvalue distribution for the bures ensemble which is shown to be broader then the quarter circle distribution characteristic of the hilbert schmidt ensemble for measures induced by partial tracing over the environment we compute exactly the two point eigenvalue correlation function

determinantal processes with number variance saturation consider dyson s hermitian brownian motion model after a finite time s where the process is started at n equidistant points on the real line these n points after time s form a determinantal process and has a limit as n this limting determinantal process has the interesting feature that it shows number variance saturation the variance of the number of particles in an interval converges to a limiting value as the length of the interval goes to infinity number variance saturation is also seen for example in the zeros of the riemann function 21 2 the process can also be constructed using non intersecting paths and we consider several variants of this construction one construction leads to a model which shows a transition from a non universal behaviour with number variance saturation to a universal sine kernel behaviour as we go up the line

the isomonodromy approach to matric models in 2d quantum gravity we consider the double scaling limit in the hermitian matrix model for 2d quantum gravity associated with the measure exp sum limits j 1 n t j z 2j n geqq 3 we show that after the appropriate modification of the contour of integration the cross migdal douglas shenker limit to the painleve i equation in the generic case of the pure gravity is valid and calculate the nonperturbative parameters of the corresponding painleve function our approach is based on the wkb analysis of the l a pair corresponding to the discrete string equation in the framework of the inverse monodromy method here we extend our results which were obtained before for the particular casesn 2 3 our analysis complements the isomonodromy approach proposed by g moore to the general string equations that come from the matrix model in the continuous limit and differ in that we apply the isomonodromy technique to investigate the double scaling limit itself

detection in multivariate non gaussian noise the applications of multivariate edgeworth series and higher order statistics to the discrete time detection of a known constant signal in multivariate non gaussian noise are considered a technique to derive suboptimum detectors from the neyman pearson optimum and locally optimum detectors is described a numerical algorithm based on knowledge of the noise cumulants is presented in order to analyze the finite sample size performance of the suboptimum detectors as an example the performance of the detectors as compared with the linear detector in multivariate gaussian gaussian mixture noise is presented via receiver operating characteristic curves numerical results indicate that the suboptimum detectors when exploiting knowledge of the dependence structure of the noise can have very good performance with respect to the linear detector

topological recursion in enumerative geometry and random matrices we review the method of symplectic invariants recently introduced to solve matrix models loop equations in the so called topological expansion and further extended beyond the context of matrix models for any given spectral curve one defines a sequence of differential forms and a sequence of complex numbers fg called symplectic invariants we recall the definition of fg s and we explain their main properties in particular symplectic invariance integrability modularity as well as their limits and their deformations then we give several examples of applications in particular matrix models enumeration of discrete surfaces maps algebraic geometry and topological strings and non intersecting brownian motions

expansions of the likelihood ratio and applications the likelihood ratio receiver although optimal is oftentimes difficult to realize to overcome this difficulty suboptimal approximations have been studied in the past one of the simplest approaches is to expand the likelihood ratio and truncate this expansion under the weak signal assumption in this paper we first study the relations between the taylor expansion and the volterra expansion under a weak signal assumption we emphasize the links between the taylor expansion with respect to the snr and the detection criteria on one hand and with robustness on the other in particular using two criteria derived from deflection we show that the minimax robust filter is the matched filter finally an application to underwater signal detection is studied showing that the degradation resulting from the use of the locally optimal receiver is small

a fast recursive least squares adaptive second order volterra filter and its performance analysis a fast recursive least squares rls adaptive nonlinear filter modeled using a second order volterra series expansion is presented the structure uses the ideas of fast rls multichannel filters and has a computational complexity of o n sup 3 multiplications where n 1 represents the memory span in number of samples of the nonlinear system model a theoretical performance analysis of its steady state behaviour in both stationary and nonstationary environments is presented the analysis shows that when the input is zero mean and gaussian distributed and the adaptive filter is operating in a stationary environment the steady state excess mean squared error due to the coefficient noise vector is independent of the statistics of the input signal the results of several simulation experiments show that the filter performs well in a variety of situations the steady state behaviour predicted by the analysis is in very good agreement with the experimental results

a note on the limiting mean distribution of singular values for products of two wishart random matrices the product of m complex random gaussian matrices of size n has recently been studied by akemann kieburg and wei they showed that for fixed m and n the joint probability distribution for the squared singular values of the product matrix forms a determinantal point process with a correlation kernel determined by certain biorthogonal polynomials that can be explicitly constructed we find that in the case m 2 the relevant biorthogonal polynomials are actually special cases of multiple orthogonal polynomials associated with macdonald functions modified bessel functions of the second kind which was first introduced by van assche and yakubovich with known results on asymptotic zero distribution of these polynomials and general theory on multiple orthogonal polynomial ensembles it is then easy to obtain an explicit expression for the distribution of squared singular values for the product of two complex random gaussian matrices in the limit of large matrix dimensions

energy correlations for a random matrix model of disordered bosons linearizing the heisenberg equations of motion around the ground state of an interacting quantum many body system one gets a time evolution generator in the positive cone of a real symplectic lie algebra the presence of disorder in the physical system determines a probability measure with support on this cone the present paper analyzes a discrete family of such measures of exponential type and does so in an attempt to capture by a simple random matrix model some generic statistical features of the characteristic frequencies of disordered bosonic quasiparticle systems the level correlation functions of the said measures are shown to be those of a determinantal process and the kernel of the process is expressed as a sum of biorthogonal polynomials while the correlations in the bulk scaling limit are in accord with sine kernel or gaussian unitary ensemble universality at the low frequency end of the spectrum an unusual type of scaling behavior is found

on knapsacks partitions and a new dynamic programming technique for trees let g be an acyclic directed graph with weights and values assigned to its vertices in the partially ordered knapsack problem we wish to find a maximum valued subset of vertices whose total weight does not exceed a given knapsack capacity and which contains every predecessor of a vertex if it contains the vertex itself we consider the special case where g is an out tree even though this special case is still np complete we observe how dynamic programming techniques can be used to construct pseudopolynomial time optimization algorithms and fully polynomial time approximation schemes for it in particular we show that a nonstandard approach we call left right dynamic programming is better suited for this problem than the standard bottom up approach and we show how this left right approach can also be adapted to the case of in trees and to a related tree partitioning problem arising in integrated circuit design we conclude by presenting complexity results which indicate that similar success cannot be expected with either problem when the restriction to trees is lifted

random matrix models with additional interactions it has been argued that despite remarkable success existing random matrix theories are not adequate at describing disordered conductors in the metallic regime due to the presence of certain two body interactions in the effective hamiltonian for the eigenvalues in addition to the standard logarithmic interaction that arises entirely from symmetry considerations we present a new method that allows exact solution of random matrix models with such additional two body interactions this should broaden the scope of random matrix models in general

large n limit of gaussian random matrices with external source part iii double scaling limit we consider the double scaling limit in the random matrix ensemble with an external source 1 over z n e n hbox tr 1 over 2 m 2 am dm defined on n n hermitian matrices where a is a diagonal matrix with two eigenvalues a of equal multiplicities the value a 1 is critical since the eigenvalues of m accumulate as n on two intervals for a 1 and on one interval for 0 a 1 these two cases were treated in parts i and ii where we showed that the local eigenvalue correlations have the universal limiting behavior known from unitary random matrix ensembles for the critical case a 1 new limiting behavior occurs which is described in terms of pearcey integrals as shown by brezin and hikami and tracy and widom we establish this result by applying the deift zhou steepest descent method to a 3 3 matrix valued riemann hilbert problem which involves the construction of a local parametrix out of pearcey integrals we resolve the main technical issue of matching the local pearcey parametrix with a global outside parametrix by modifying an underlying riemann surface

tracy widom limit for the largest eigenvalue of a large class of complex sample covariance matrices we consider the asymptotic fluctuation behavior of the largest eigenvalue of certain sample covariance matrices in the asymptotic regime where both dimensions of the corresponding data matrix go to infinity more precisely let x be an n x p matrix and let its rows be i i d complex normal vectors with mean 0 and covariance p we show that for a large class of covariance matrices p the largest eigenvalue of x x is asymptotically distributed after recentering and rescaling as the tracy widom distribution that appears in the study of the gaussian unitary ensemble we give explicit formulas for the centering and scaling sequences that are easy to implement and involve only the spectral distribution of the population covariance n and p the main theorem applies to a number of covariance models found in applications for example well behaved toeplitz matrices as well as covariance matrices whose spectral distribution is a sum of atoms under some conditions on the mass of the atoms are among the models the theorem can handle generalizations of the theorem to certain spiked versions of our models and a s results about the largest eigenvalue are given we also discuss a simple corollary that does not require normality of the entries of the data matrix and some consequences for applications in multivariate statistics

a pegging approach to the precedence constrained knapsack problem the knapsack problem kp is generalized to the case where items are partially ordered through a set of precedence relations as in ordinary kps each item is associated with profit and weight the knapsack has a fixed capacity and the problem is to determine the set of items to be packed in the knapsack however each item can be accepted only when all the preceding items have been included in the knapsack the knapsack problem with these additional constraints is referred to as the precedence constrained knapsack problem pckp to solve pckp exactly we present a pegging approach where the size of the original problem is reduced by applying the lagrangian relaxation followed by a pegging test through this approach we are able to solve pckps with thousands of items within a few minutes on an ordinary workstation

universality for the pearcey process consider non intersecting brownian motions on the line leaving from the origin and forced to two arbitrary points letting the number of brownian particles tend to infinity and upon rescaling there is a point of bifurcation where the support of the density of particles goes from one interval to two intervals in this paper we show that at that very point of bifurcation a cusp appears near which the brownian paths fluctuate like the pearcey process this is a universality result within this class of problems tracy and widom obtained such a result in the symmetric case when the two target points are symmetric with regard to the origin this asymmetry enabled us to improve considerably a result concerning the non linear partial differential equations governing the transition probabilities for the pearcey process obtained by adler and van moerbeke c 2010 elsevier b v all rights reserved

level spacing of random matrices in an external source in an earlier work we had considered a gaussian ensemble of random matrices in the presence of a given external matrix source the measure is no longer unitary invariant and the usual techniques based on orthogonal polynomials or on the coulomb gas representation are not available nevertheless the n point correlation functions are still given in terms of the determinant of a kernel known through an explicit integral representation this kernel is no longer symmetric though and is not readily accessible to standard methods in particular finding the level spacing probability is always a delicate problem in fredholm theory and we have to reconsider the problem within our model we find a new class of universality for the level spacing distribution when the spectrum of the source is ajusted to produce a vanishing gap in the density of the state the problem is solved through coupled non linear differential equations which turn out to form a hamiltonian system as a result we find that the level spacing probability p s behaves like exp c s 8 over 3 for large spacing s this is consistent with the asymptotic behavior exp c s 2 beta 2 whenever the density of state behaves near the edge as rho lambda sim lambda beta

pdes for the gaussian ensemble with external source and the pearcey distribution the present paper studies a gaussian hermitian random matrix ensemble with external source given by a fixed diagonal matrix with two eigenvalues a as a first result the probability that the eigenvalues of the ensemble belong to an interval e satisfies a fourth order pde with quartic nonlinearity the variables are the eigenvalue a and the boundary of e this equation enables one to find a pde for the pearcey distribution the latter describes the statistics of the eigenvalues near the closure of a gap i e when the support of the equilibrium measure for large size random matrices has a gap that can be made to close the gaussian hermitian random matrix ensemble with external source described above has this feature the pearcey distribution is shown to satisfy a fourth order pde with cubic nonlinearity this also gives the pde for the transition probability of the pearcey process a limiting process associated with nonintersecting brownian motions on r c 2006 wiley periodicals inc

random matrices with equispaced external source we study hermitian random matrix models with an external source matrix which has equispaced eigenvalues and with an external field such that the limiting mean density of eigenvalues is supported on a single interval as the dimension tends to infinity we obtain strong asymptotics for the multiple orthogonal polynomials associated to these models and as a consequence for the average characteristic polynomials one feature of the multiple orthogonal polynomials analyzed in this paper is that the number of orthogonality weights of the polynomials grows with the degree nevertheless we are able to characterize them in terms of a pair of 2 1 vector valued riemann hilbert problems and to perform an asymptotic analysis of the riemann hilbert problems

large n limit of gaussian random matrices with external source part i we continue the study of the hermitian random matrix ensemble with external source open image in new window where a has two distinct eigenvalues a of equal multiplicity this model exhibits a phase transition for the value a 1 since the eigenvalues of m accumulate on two intervals for a 1 and on one interval for 0 1 was treated in part i where it was proved that local eigenvalue correlations have the universal limiting behavior which is known for unitarily invariant random matrices that is limiting eigenvalue correlations are expressed in terms of the sine kernel in the bulk of the spectrum and in terms of the airy kernel at the edge in this paper we establish the same results for the case 0 a 1 as in part i we apply the deift zhou steepest descent analysis to a 3 3 matrix riemann hilbert problem due to the different structure of an underlying riemann surface the analysis includes an additional step involving a global opening of lenses which is a new phenomenon in the steepest descent analysis of riemann hilbert problems

application of the volterra functional expansion in the detection of nonlinear functions of gaussian processes detection of a memoryless nonlinear functional of a gaussian process in additive gaussian white noise is considered the volterra functional expansion for the likelihood ratio and two examples of calculating the kernels are presented it is shown that kernels up to third order can be obtained for a hard limited gaussian process and for the absolute value of a gaussian process for the case of hard limiting the kernels are nonlinear functions of the autocorrelation of the gaussian process for the absolute value case the kernels are nonlinear functions of the kernel derived for the linear problem a monte carlo simulation of receiver performance is presented for the case of detection of the absolute value of a first order butterworth process in additive gaussian white noise the suboptimal detector is obtained by truncating the log likelihood ratio to second order

large deviations for disordered bosons and multiple orthogonal polynomial ensembles we prove a large deviations principle for the empirical measures of a class of biorthogonal and multiple orthogonal polynomial ensembles that includes biorthogonal laguerre jacobi and hermite ensembles the matrix model of lueck sommers and zirnbauer for disordered bosons the stieltjes wigert matrix model of chern simons theory and angelesco ensembles

on wright s generalized bessel kernel abstract in this paper we consider the wright s generalized bessel kernel k x y defined by x 0 1 j 1 1 u x j 1 u y u d u 1 0 where j a b x j 0 x j j a b j a c b 1 is wright s generalization of the bessel function this non symmetric kernel which generalizes the classical bessel kernel corresponding to 1 in random matrix theory is the hard edge scaling limit of the correlation kernel for certain muttalib borodin ensembles we show that if is rational i e m n with m n n g c d m n 1 and m 1 m n the wright s generalized bessel kernel is integrable in the sense of its izergin korepin slavnov we then come to the fredholm determinant of this kernel over the union of several scaled intervals which can also be interpreted as the gap probability the probability of finding no particles on these intervals the integrable structure allows us to obtain a system of coupled partial differential equations associated with the corresponding fredholm determinant as well as a hamiltonian interpretation as a consequence we are able to represent the gap probability over a single interval 0 s in terms of a solution of a system of nonlinear ordinary differential equations

clique based facets for the precedence constrained knapsack problem we consider a knapsack problem with precedence constraints imposed on pairs of items known as the precedence constrained knapsack problem pckp this problem has applications in manufacturing and mining and also appears as a subproblem in decomposition techniques for network design and related problems we present a new approach for determining facets of the pckp polyhedron based on clique inequalities a comparison with existing techniques that lift knapsack cover inequalities for the pckp is also presented it is shown that the clique based approach generates facets that cannot be found through the existing cover based approaches and that the addition of clique based inequalities for the pckp can be computationally beneficial for both pckp instances arising in real applications and applications in which pckp appears as an embedded structure

zooming in on local level statistics by supersymmetric extension of free probability we consider unitary ensembles of hermitian n n matrices governed by a confining potential nv with analytic and uniformly convex v from earlier work it is known that the large n limit of the characteristic function for a finite rank fourier variable k is determined by the voiculescu r transform a key object in free probability theory going beyond these results we argue that the same holds true when the finite rank operator k has the form that is required by the wegner efetov supersymmetry method this insight leads to a potent new technique for the study of local statistics e g level correlations we illustrate the new technique by demonstrating universality in a random matrix model of stochastic scattering