violation of the fluctuation dissipation theorem in glassy systems basic notions and the numerical evidence this review reports on the research done during past years on violations of the fluctuation dissipation theorem fdt in glassy systems it is focused on the existence of a quasi fluctuation dissipation theorem qfdt in glassy systems and the current supporting knowledge gained from numerical simulation studies it covers a broad range of non stationary aging and stationary driven systems such as structural glasses spin glasses coarsening systems ferromagnetic models at criticality trap models models with entropy barriers kinetically constrained models sheared systems and granular media the review is divided into four main parts 1 an introductory section explaining basic notions related to the existence of the fdt in equilibrium and its possible extension to the glassy regime qfdt 2 a description of the basic analytical tools and results derived in the framework of some exactly solvable models 3 a detailed report of the current evidence in favour of the qfdt and 4 a brief digression on the experimental evidence in its favour this review is intended for inexpert readers who want to learn about the basic notions and concepts related to the existence of the qfdt as well as for the more expert readers who may be interested in more specific results

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

universality at the edge of the spectrum for unitary orthogonal and symplectic ensembles of random matrices we prove universality at the edge of the spectrum for unitary 2 orthogonal 1 and symplectic 4 ensembles of random matrices in the scaling limit for a class of weights w x e v x where v is a polynomial v x 2mx2m 2m 0 the precise statement of our results is given in theorem 1 1 and corollaries 1 2 and 1 4 below for the same class of weights a proof of universality in the bulk of the spectrum is given in 12 for the unitary ensembles and in 9 for the orthogonal and symplectic ensembles our starting point in the unitary case is 12 and for the orthogonal and symplectic cases we rely on our recent work 9 which in turn depends on the earlier work of widom 46 and tracy and widom 42 as in 9 the uniform plancherel rotach type asymptotics for the orthogonal polynomials found in 12 plays a central role the formulae in 46 express the correlation kernels for 1 4 as a sum of a christoffel darboux cd term as in the case 2 together with a correction term in the bulk scaling limit 9 the correction term is of lower order and does not contribute to the limiting form of the correlation kernel by contrast in the edge scaling limit considered here the cd term and the correction term contribute to the same order this leads to additional technical difficulties over and above 49 2006 wiley periodicals inc

some universal properties for restricted trace gaussian orthogonal unitary and symplectic ensembles consider fixed and bounded trace gaussian orthogonal unitary and symplectic ensembles closely related to gaussian ensembles without any constraint for three restricted trace gaussian ensembles we prove universal limits of correlation functions at zero and at the edge of the spectrum edge our argument also applies to restricted trace ensembles with monomial potentials in addition by using the universal result in the bulk for fixed trace gaussian unitary ensemble which has been obtained by gotze and gordin we also prove the universal limits of correlation functions everywhere in the bulk for bounded trace gaussian unitary ensemble

multiparametric quantum gl 2 lie bialgebras quantum r matrices and non relativistic limits multiparametric quantum deformations of gl 2 are studied through a complete classification of gl 2 lie bialgebra structures from these the non relativistic limit leading to harmonic oscillator lie bialgebras is implemented by means of a contraction procedure new quantum deformations of gl 2 together with their associated quantum r matrices are obtained and other known quantizations are recovered and classified several connections with integrable models are outlined

level spacings distribution for large random matrices gaussian fluctuations we study the level spacings distribution for eigenvalues of large n x n matrices from the classical compact groups in the scaling limit when the mean distance between nearest eigenvalues equals 1 defining by 77n s the number of nearest neighbors spacings greater than s 0 smaller than s 0 we prove functional limit theorem for the process 77n s e n s n1 2 giving weak convergence of this distribution to some gaussian random process on 0 x the limiting gaussian random process is universal for all classical compact groups it is holder continuous with any exponent less than 1 2 similar results can be obtained for the n level spacings distribution

generalized christoffel darboux formula for classical skew orthogonal polynomials we show that skew orthogonal functions defined with respect to jacobi weight wa b x 1 x a 1 x b a b 1 including the limiting cases of laguerre wa x xa e x a 1 and gaussian weight satisfy three term recursion relation in the quaternion space from this we derive generalized christoffel darboux gcd formulae for kernel functions arising in the study of the corresponding orthogonal and symplectic ensembles of random 2n 2n matrices using the gcd formulae we calculate the level densities and prove that in the bulk of the spectrum under appropriate scaling the eigenvalue correlations are universal we also provide evidence to show that there exists a mapping between skew orthogonal functions arising in the study of orthogonal and symplectic ensembles of random matrices

deformed laguerre hahn orthogonal polynomials on the real line abstract one derives discrete dynamical systems related to laguerre hahn orthogonal polynomials one studies deformations of the recurrence relation coefficients of the orthogonal polynomials under a t dependence on the coefficients of the riccati differential equation for the related stieltjes function

scale space and edge detection using anisotropic diffusion a new definition of scale space is suggested and a class of algorithms used to realize a diffusion process is introduced the diffusion coefficient is chosen to vary spatially in such a way as to encourage intraregion smoothing rather than interregion smoothing it is shown that the no new maxima should be generated at coarse scales property of conventional scale space is preserved as the region boundaries in the approach remain sharp a high quality edge detector which successfully exploits global information is obtained experimental results are shown on a number of images parallel hardware implementations are made feasible because the algorithm involves elementary local operations replicated over the image

painleve equations nonlinear special functions the six painleve equations pi pvi were first discovered about a hundred years ago by painleve and his colleagues in an investigation of nonlinear second order ordinary differential equations recently there has been considerable interest in the painleve equations primarily due to the fact that they arise as reductions of the soliton equations which are solvable by inverse scattering consequently the painleve equations can be regarded as completely integrable equations and possess solutions which can be expressed in terms of solutions of linear integral equations despite being nonlinear equations although first discovered from strictly mathematical considerations the painleve equations have arisen in a variety of important physical applications including statistical mechanics plasma physics nonlinear waves quantum gravity quantum field theory general relativity nonlinear optics and fibre optics the painleve equations may be thought of a nonlinear analogues of the classical special functions they possess hierarchies of rational solutions and one parameter families of solutions expressible in terms of the classical special functions for special values of the parameters further the painleve equations admit symmetries under affine weyl groups which are related to the associated backlund transformations in this paper i discuss some of the remarkable properties which the painleve equations possess including connection formulae backlund transformations associated discrete equations and hierarchies of exact solutions in particular the second painleve equation pii is used to illustrate these properties and some of the applications of pii are also discussed

propagation separation approach for local likelihood estimation the paper presents a unified approach to local likelihood estimation for a broad class of nonparametric models including e g the regression density poisson and binary response model the method extends the adaptive weights smoothing aws procedure introduced in polzehl and spokoiny 2000 in context of image denoising the main idea of the method is to describe a greatest possible local neighborhood of every design point x i in which the local parametric assumption is justified by the data the method is especially powerful for model functions having large homogeneous regions and sharp discontinuities the performance of the proposed procedure is illustrated by numerical examples for density estimation and classification we also establish some remarkable theoretical nonasymptotic results on properties of the new algorithm this includes the propagation property which particularly yields the root n consistency of the resulting estimate in the homogeneous case we also state an oracle result which implies rate optimality of the estimate under usual smoothness conditions and a separation result which explains the sensitivity of the method to structural changes

a method of linearization for painleve equations painleve iv v abstract a method to linearize the initial value problem of the painleve equations iv v is given the procedure involves formulating a riemann hilbert boundary value problem on intersecting lines for the inverse monodromy problem this boundary value problem is reduced to a sequence of standard problems on single lines in a certain range of parameter space schlesinger transformations allow one to completely cover the parameter space special solutions are constructed from special cases of the riemann problem as well

averages of characteristic polynomials in random matrix theory we compute averages of products and ratios of characteristic polynomials associated with orthogonal unitary and symplectic ensembles of random matrix theory the pfaffian determinantal formulae for these averages are obtained and the bulk scaling asymptotic limits are found for ensembles with gaussian weights classical results for the correlation functions of the random matrix ensembles and their bulk scaling limits are deduced from these formulae by a simple computation we employ a discrete approximation method the problem is solved for discrete analogues of random matrix ensembles originating from representation theory and then a limit transition is performed exact pfaffian determinantal formulae for the discrete averages are proven using standard tools of linear algebra no application of orthogonal or skew orthogonal polynomials is needed 2005 wiley periodicals inc

skew orthogonal polynomials differential systems and random matrix theory we study skew orthogonal polynomials with respect to the weight function exp 2v x with v x 2dk 1 uk k xk u2d 0 d 0 a finite subsequence of such skew orthogonal polynomials arising in the study of orthogonal and symplectic ensembles of random matrices satisfies a system of differential difference deformation equation the vectors formed by such subsequence have the rank equal to the degree of the potential in the quaternion sense these solutions satisfy certain compatibility condition and hence admit a simultaneous fundamental system of solutions

a riemannian framework for tensor computing tensors are nowadays a common source of geometric information in this paper we propose to endow the tensor space with an affine invariant riemannian metric we demonstrate that it leads to strong theoretical properties the cone of positive definite symmetric matrices is replaced by a regular and complete manifold without boundaries null eigenvalues are at the infinity the geodesic between two tensors and the mean of a set of tensors are uniquely defined etc we have previously shown that the riemannian metric provides a powerful framework for generalizing statistics to manifolds in this paper we show that it is also possible to generalize to tensor fields many important geometric data processing algorithms such as interpolation filtering diffusion and restoration of missing data for instance most interpolation and gaussian filtering schemes can be tackled efficiently through a weighted mean computation linear and anisotropic diffusion schemes can be adapted to our riemannian framework through partial differential evolution equations provided that the metric of the tensor space is taken into account for that purpose we provide intrinsic numerical schemes to compute the gradient and laplace beltrami operators finally to enforce the fidelity to the data either sparsely distributed tensors or complete tensors fields we propose least squares criteria based on our invariant riemannian distance which are particularly simple and efficient to solve

the fixed trace β hermite ensemble of random matrices and the low temperature distribution of the determinant of an n n β hermite matrix the hermite ensemble he of tridiagonal n n random matrices of dumitriu and edelman 2002 j math phys 43 5830 is a continuum of ensembles in which the reciprocal of the temperature in the 2d electrostatic interpretation of the eigenvalue characteristics can take any value the eigenvalue distributions coincide with those of the classical gaussian ensembles goe gue gse for 1 2 4 a fixed trace hermite ensemble fthe is defined from the he and is used to extend the spherical ensembles of classical symmetries to spherical ensembles at low temperature when for a fixed value of n the asymptotic distributions of reduced determinants dn of random n n h and fth matrices are shown to be standard gaussians accordingly the fluctuations of the potential at the origin ln dn have a generalized gumbel distribution at low temperature for large n and large a ln n variance results from the strongly correlated fluctuations of eigenvalues around their equilibrium positions

multiscale adaptive regression models for neuroimaging data neuroimaging studies aim to analyze imaging data with complex spatial patterns in a large number of locations called voxels on a two dimensional 2d surface or in a 3d volume conventional analyses of imaging data include two sequential steps spatially smoothing imaging data and then independently fitting a statistical model at each voxel however conventional analyses suffer from the same amount of smoothing throughout the whole image the arbitrary choice of smoothing extent and low statistical power in detecting spatial patterns we propose a multiscale adaptive regression model marm to integrate the propagation separation ps approach polzehl and spokoiny 2000 2006 with statistical modeling at each voxel for spatial and adaptive analysis of neuroimaging data from multiple subjects marm has three features being spatial being hierarchical and being adaptive we use a multiscale adaptive estimation and testing procedure maet to utilize imaging observations from the neighboring voxels of the current voxel to adaptively calculate parameter estimates and test statistics theoretically we establish consistency and asymptotic normality of the adaptive parameter estimates and the asymptotic distribution of the adaptive test statistics our simulation studies and real data analysis confirm that marm significantly outperforms conventional analyses of imaging data

nonlinear susceptibilities of spherical models the static and dynamic susceptibilities for a general class of mean field random orthogonal spherical spin glass models are studied we show how the static and dynamical properties of the linear and nonlinear susceptibilities depend on the behaviour of the density of states of the two body interaction matrix in the neighbourhood of the largest eigenvalue our results are compared with experimental results and also with those of the droplet theory of spin glasses

an exact formula for general spectral correlation function of random hermitian matrices we have found an exact formula expressing a general correlation function containing both products and ratios of characteristic polynomials of random hermitian matrices the answer is given in the form of a determinant an essential difference from the previously studied correlation functions of products only is the appearance of non polynomial functions along with the orthogonal polynomials these non polynomial functions are the cauchy transforms of the orthogonal polynomials the result is valid for arbitrary ensemble of 2 symmetry class and generalizes recent asymptotic formulae obtained for gaussian unitary ensemble and its chiral counterpart by different methods

differential systems for biorthogonal polynomials appearing in 2 matrix models and the associated riemann hilbert problem we consider biorthogonal polynomials that arise in the study of a generalization of two matrix hermitian models with two polynomial potentials v1 x v2 y of any degree with arbitrary complex coefficients finite consecutive subsequences of biorthogonal polynomials windows of lengths equal to the degrees of the potentials v1 and v2 satisfy systems of ode s with polynomial coefficients as well as pde s deformation equations with respect to the coefficients of the potentials and recursion relations connecting consecutive windows a compatible sequence of fundamental systems of solutions is constructed for these equations the stokes sectorial asymptotics of these fundamental systems are derived through saddle point integration and the riemann hilbert problem characterizing the differential equations is deduced

weight vectors of the basic a1 1 module and the littlewood richardson rule the basic representation of a1 1 is studied the weight vectors are represented in terms of schur functions a suitable base of any weight space is given the littlewood richardson rule appears in the linear relations among weight vectors

large n expansions and painleve hierarchies in the hermitian matrix model we present a method to characterize and compute the large n formal asymptotics of regular and critical hermitian matrix models with general even potentials in the one cut and two cut cases our analysis is based on a method to solve continuum limits of the discrete string equation which uses the resolvent of the lax operator of the underlying toda hierarchy this method also leads to an explicit formulation in terms of coupling constants and critical parameters of the members of the painleve i and painleve ii hierarchies associated with one cut and two cut critical models respectively

schur functions and affine lie algebras abstract we make use of the representation theory of the infinite dimensional lie algebrasa b and formula to derive explicit formulas relating schur sp functions to schur ss functions

asymptotics of the largest zeros of some orthogonal polynomials we study the asymptotics of the largest zeros of the wilson hermite and q laguerre polynomials using two distinct techniques the first is based on the coulomb fluid technique developed in a previous paper where the primary input is the weight function while the second uses the method of chain sequences which supplies inequalities for the largest zeros using the recurrence coefficients we also investigate the asymptotics of the largest zeros of the polynomials orthogonal to the weight functions for c 0 and m a positive even integer

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

clinical dt mri estimation smoothing and fiber tracking with log euclidean metrics diffusion tensor magnetic resonance imaging dt mri or dti is an imaging modality that is gaining importance in clinical applications however in a clinical environment data have to be acquired rapidly often at the expense of the image quality this often results in dti datasets that are not suitable for complex postprocessing like fiber tracking we propose a new variational framework to improve the estimation of dt mri in this clinical context most of the existing estimation methods rely on a log gaussian noise gaussian noise on the image logarithms or a gaussian noise that do not reflect the rician nature of the noise in mr images with a low signal to noise ratio snr with these methods the rician noise induces a shrinking effect the tensor volume is underestimated when other noise models are used for the estimation in this paper we propose a maximum likelihood strategy that fully exploits the assumption of a rician noise to further reduce the influence of the noise we optimally exploit the spatial correlation by coupling the estimation with an anisotropic prior previously proposed on the spatial regularity of the tensor field itself which results in a maximum a posteriori estimation optimizing such a nonlinear criterion requires adapted tools for tensor computing we show that riemannian metrics for tensors and more specifically the log euclidean metrics are a good candidate and that this criterion can be efficiently optimized experiments on synthetic data show that our method correctly handles the shrinking effect even with very low snr and that the positive definiteness of tensors is always ensured results on real clinical data demonstrate the truthfulness of the proposed approach and show promising improvements of fiber tracking in the brain and the spinal cord

eigenvalue density in hermitian matrix models by the lax pair method in this paper a new method is discussed to derive the eigenvalue density in a hermitian matrix model with a general potential the density is considered on one interval or multiple disjoint intervals the method is based on lax pair theory and the cayley hamilton theorem by studying the orthogonal polynomials associated with the hermitian matrix model it is obtained that the restriction conditions for the parameters in the density are connected to the discrete painleve i equation and the results are related to the scalar riemann hilbert problem some special density functions are also discussed in association with the known results in this subject

gaussian fluctuation for the number of particles in airy bessel sine and other determinantal random point fields we prove the central limit theorem clt for the number of eigenvalues near the spectrum edge for certain hermitian ensembles of random matrices to derive our results we use a general theorem essentially due to costin and lebowitz concerning the gaussian fluctuation of the number of particles in random point fields with determinantal correlation functions as another corollary of the costin lebowitz theorem we prove the clt for the empirical distribution function of the eigenvalues of random matrices from classical compact groups

spatial shrinkage estimation of diffusion tensors on diffusion weighted imaging data diffusion tensor imaging dti based on the diffusion weighted imaging dwi data acquired from magnetic resonance experiments has been widely used to analyze the physical structure of white matter fibers in the human brain in vivo the raw dwi data however carry noise this contaminates the diffusion tensor dt estimates and introduces systematic bias into the induced eigenvalues these bias components affect the effectiveness of fiber tracking algorithms in this article we propose a two stage spatial shrinkage estimation spske procedure to accommodate the spatial information carried in dwi data in dt estimation and to reduce the bias components in the corresponding derived eigenvalues to this end in the framework of the heteroscedastic linear model spske incorporates l 1 type penalization and the locally weighted least square function the theoretical properties of spske are explored the effectiveness of spske is further illustrated by simulation and real data examples supplementary materi

basic representations of a2l 2 and dl 1 2 and the polynomial solutions to the reduced bkp hierarchies basic representations of a2l 2 and dl 1 2 are studied the weight vectors are represented in terms of schur s q functions the method to get the polynomial solutions to the reduced bkp hierarchies is shown to be equivalent to a certain rule in the maya game

on external fields created by fixed charges abstract in this paper equilibrium measures in the presence of external fields created by fixed charges are analyzed these external fields are a particular case of the so called rational external fields in the sense that their derivatives are rational functions along with some general results a thorough analysis of the particular case of two fixed negative charges attractors is presented indeed the main result of the paper deals with this particular case as for the main tools used this paper is a natural continuation of 31 where polynomial external fields were thoroughly studied and 37 where rational external fields with a polynomial part were considered however the absence of the polynomial part in the external fields analyzed in the current paper adds a considerable difficulty to solve the problem and justifies its separated treatment moreover it is noteworthy to point out the simplicity and beauty of the results obtained

phase structure and asymptotic zero densities of orthogonal polynomials in the cubic model we apply the method we have described in a previous paper 2013 to determine the phase structure of asymptotic zero densities of the standard cubic model of non hermitian orthogonal polynomials we provide a complete description of the two phases the one cut phase and the two cut phase and analyze the phase transition processes of the types splitting of a cut birth and death of a cut

universality of the double scaling limit in random matrix models we study unitary random matrix ensembles in the critical case where the limiting mean eigenvalue density vanishes quadratically at an interior point of the support we establish universality of the limits of the eigenvalue correlation kernel at such a critical point in a double scaling limit the limiting kernels are constructed out of functions associated with the second painleve equation this extends a result of bleher and its for the special case of a critical quartic potential the two main tools we use are equilibrium measures and riemann hilbert problems in our treatment of equilibrium measures we allow a negative density near the critical point which enables us to treat all cases simultaneously the asymptotic analysis of the riemann hilbert problem is done with the deift zhou steepest descent analysis for the construction of a local parametrix at the critical point we introduce a modification of the approach of baik deift and johansson so that we are able to satisfy the required jump properties exactly c 2005 wiley periodicals inc

semiclassical expansions in the toda hierarchy and the hermitian matrix model an iterative algorithm for determining a class of solutions of the dispersionful 2 toda hierarchy characterized by string equations is developed this class includes the solution which underlies the large n limit of the hermitian matrix model in the one cut case it is also shown how the double scaling limit can be naturally formulated in this scheme

generic behavior of the density of states in random matrix theory and equilibrium problems in the presence of real analytic external fields the equilibrium measure in the presence of an external field plays a role in a number of areas in analysis for example in random matrix theory the limiting mean density of eigenvalues is precisely the density of the equilibrium measure typical behavior for the equilibrium measure is 1 it is positive on the interior of a finite number of intervals 2 it vanishes like a square root at endpoints and 3 outside the support there is strict inequality in the euler lagrange variational conditions if these conditions hold then the limiting local eigenvalue statistics is loosely described by a bulk in which there is universal behavior involving the sine kernel and edge effects in which there is a universal behavior involving the airy kernel through techniques from potential theory and integrable systems we show that this regular behavior is generic for equilibrium measures associated with real analytic external fields in particular we show that for any one parameter family of external fields v c the equilibrium measure exhibits this regular behavior except for an at most countable number of values of c we discuss applications of our results to random matrices orthogonal polynomials and integrable systems 2000 john wiley sons inc

multi critical unitary random matrix ensembles and the general painleve ii equation we study unitary random matrix ensembles of the form z n detm 2 e ntrvwdm where a 1 2 and v is such that the limiting mean eigenvalue density for n n oc and n n 1 vanishes quadratically at the origin in order to compute the double scaling limits of the eigenvalue correlation kernel near the origin we use the deift zhou steepest descent method applied to the riemann hilbert problem for orthogonal polynomials on the real line with respect to the weight x 2ae nv x here the main focus is on the construction of a local parametrix near the origin with functions associated with a special solution qa of the painleve ii equation q sq 2q3 a we show that qa has no real poles for a 1 2 by proving the solvability of the corresponding riemann hilbert problem we also show that the asymptotics of the recurrence coefficients of the orthogonal polynomials can be expressed in terms of qa in the double scaling limit

limiting laws of linear eigenvalue statistics for hermitian matrix models we study the variance and the laplace transform of the probability law of linear eigenvalue statistics of unitary invariant matrix models of n n hermitian matrices as n assuming that the test function of statistics is smooth enough and using the asymptotic formulas by deift et al commun pure appl math 52 1325 1425 1999 for orthogonal polynomials with varying weights we show first that if the support of the density of states of the model consists of q 2 intervals then in the global regime the variance of statistics is a quasiperiodic function of n as n generically in the potential determining the model we show next that the exponent of the laplace transform of the probability law is not in general 1 2 variance as it should be if the central limit theorem would be valid and we find the asymptotic form of the laplace transform of the probability law in certain cases

a riemann hilbert problem for biorthogonal polynomials we characterize the biorthogonal polynomials that appear in the theory of coupled random matrices via a riemann hilbert problem our riemann hilbert problem is different from the ones that were proposed recently by ercolani and mclaughlin kapaev and bertola et al we believe that our formulation may be tractable to asymptotic analysis

exact densities of states of fixed trace ensembles of random matrices the densities of states and the associated characteristic functions of fixed trace ensembles ftes of n n random matrices mn tr mn mn constant are calculated exactly at finite n in the case of real symmetric matrices and of hermitian matrices the exact radial density is calculated at finite n for fixed trace ensembles of complex matrices with no further restrictions on the entries the density calculated in the hermitian case coincides with very recent literature results the exact finite n density of states of any ensemble of n n random matrices of a given symmetry whose probability density depends only on tr sn sn is simply obtained from the density of the fte of the same symmetry by a one dimensional integral

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

universal results for correlations of characteristic polynomials riemann hilbert approach we prove that general correlation functions of both ratios and products of characteristic polynomials of hermitian random matrices are governed by integrable kernels of three different types a those constructed from orthogonal polynomials b those constructed from cauchy transforms of the same orthogonal polynomials and finally c those constructed from both orthogonal polynomials and their cauchy transforms these kernels are related with the riemann hilbert problem for orthogonal polynomials for the correlation functions we obtain exact expressions in the form of determinants of these kernels derived representations enable us to study asymptotics of correlation functions of characteristic polynomials via the deift zhou steepest descent stationary phase method for riemann hilbert problems and in particular to find negative moments of characteristic polynomials this reveals the universal parts of the correlation functions and moments of characteristic polynomials for an arbitrary invariant ensemble of 2 symmetry class

test of significance based on wavelet thresholding and neyman s truncation abstract traditional nonparametric tests such as the kolmogorov smirnov test and the cramer von mises test are based on the empirical distribution functions although these tests possess root n consistency they effectively use only information contained in the low frequencies this leads to low power in detecting fine features such as sharp and short aberrants as well as global features such as high frequency alternations the drawback can be repaired via smoothing based test statistics in this article we propose two such kind of test statistics based on the wavelet thresholding and the neyman truncation we provide extensive evidence to demonstrate that the proposed tests have higher power in detecting sharp peaks and high frequency alternations while maintaining the same capability in detecting smooth alternative densities as the traditional tests similar conclusions can be made for two sample nonparametric tests of distribution functions in that case the traditional linear rank tests such as th

limit theorems for biorthogonal ensembles and related combinatorial identities abstract we study the fluctuations of certain biorthogonal ensembles for which the underlying family p q satisfies a finite term recurrence relation of the form x p x j p x for polynomial linear statistics of such ensembles we reformulate the cumulant method introduced in 53 in terms of counting certain lattice paths on the adjacency graph of the recurrence matrix j in the spirit of 12 we show that the asymptotic fluctuations are described by the right limits of the matrix j moreover whenever the right limit is a laurent matrix we show that the clt is equivalent to soshnikov s main combinatorial lemma we discuss several applications to unitary invariant hermitian random matrices in particular we provide a general central limit theorem clt and a law of large numbers in the one cut regime we also prove a clt for the square singular values of the product of independent complex rectangular ginibre matrices as well as for the laguerre and jacobi biorthogonal ensembles introduced in 7 and we explain how to recover the equilibrium measure from the asymptotics of the recurrence coefficients finally we discuss the connection with the strong szeg limit theorem where this combinatorial method originates

eigenvector distribution of wigner matrices we consider n n hermitian or symmetric random matrices with independent entries the distribution of the i j th matrix element is given by a probability measure ij whose first two moments coincide with those of the corresponding gaussian ensemble we prove that the joint probability distribution of the components of eigenvectors associated with eigenvalues close to the spectral edge agrees with that of the corresponding gaussian ensemble for eigenvectors associated with bulk eigenvalues the same conclusion holds provided the first four moments of the distribution ij coincide with those of the corresponding gaussian ensemble more generally we prove that the joint eigenvector eigenvalue distributions near the spectral edge of two generalized wigner ensembles agree provided that the first two moments of the entries match and that one of the ensembles satisfies a level repulsion estimate if in addition the first four moments match then this result holds also in the bulk

double scaling limit in the random matrix model the riemann hilbert approach we prove the existence of the double scaling limit in the unitary matrix model with quartic interaction and we show that the correlation functions in the double scaling limit are expressed in terms of the integrable kernel determined by the function for the hastings mcleod solution to the painleve ii equation the proof is based on the riemann hilbert approach and the central point of the proof is an analysis of analytic semiclassical asymptotics for the function at the critical point in the presence of four coalescing turning points 2003 wiley periodicals inc

universality at the edge of the spectrum in wigner random matrices we prove universality at the edge for rescaled correlation functions of wigner random matrices in the limit n as a corollary we show that after proper rescaling the 1th 2nd 3rd etc eigenvalues of wigner random hermitian resp real symmetric matrix weakly converge to the distributions established by tracy and widom in g u e g o e cases

deformed minkowski spaces classification and properties using general but simple covariance arguments we classify the quantum minkowski spaces for dimensionless deformation parameters this requires a previous analysis of the associated lorentz groups which reproduces a previous classification by woronowicz and zakrzewski as a consequence of the unified analysis presented we give the commutation properties the deformed and central length element and the metric tensor for the different spacetime algebras

universality of general beta ensembles we prove the universality of the ensembles with convex analytic potentials and for any 0 i e we show that the spacing distributions of log gases at any inverse temperature coincide with those of the gaussian ensembles

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

bulk eigenvalue statistics for random regular graphs ak was partly supported by swiss national science foundation grant 144662 hty was partly supported by nsf grant dms 1307444 and a simons investigator fellowship

universality for eigenvalue correlations from the unitary ensemble associated with a family of singular weights we study the asymptotic behavior of the eigenvalue correlations for the unitary ensemble associated with a family of singular weights w x exp 1 x2 x 1 1 for 0 when 0 1 2 these are szego class weights and are non szego when 1 2 it is proved that the behavior in the bulk of the spectrum is described in terms of the sine kernel which persists the so called universality results while the local behavior at the edge of the spectrum is described in terms of the airy kernel a specific scaling of the limit reflects the singular behavior of orthogonal polynomials on 1 1 with respect to the weight w x

arbitrary unitarily invariant random matrix ensembles and supersymmetry we generalize the supersymmetry method in random matrix theory to ensembles which are unitarily invariant but otherwise arbitrary our exact approach extends a previous contribution in which we constructed a supersymmetric representation for the class of norm dependent random matrix ensembles here we derive a supersymmetric formulation under very general circumstances a reduced probability density and a projector are identified that map the probability density from ordinary to superspace furthermore it is demonstrated that setting up the theory in fourier superspace has considerable advantages general and exact expressions for the correlation functions are given we also show how the use of hyperbolic symmetry can be circumvented in the present context in which the nonlinear model is not used we construct exact supersymmetric integral representations of the correlation functions for arbitrary positions of the imaginary increments in the green s functions

random words toeplitz determinants and integrable systems ii this paper connects the analysis of the length of the longest weakly increasing subsequence of inhomogeneous random words to a riemnn hilbert problem and an associated system of integrable partial differential equations in particular we show that the poissonization of the distribution function of this length can be identified as the jimbo miwa ueno tau function

exactly soluble isotropic spin ladder models the undeformed limit of the dilute two colour braid monoid algebra gives a natural basis for the description of spin ladder models and allows different baxterization ansatze based on this observation we find the complete classification of exactly soluble generalized isotropic spin ladder models

critical edge behavior in the modified jacobi ensemble and the painleve v transcendents we study the jacobi unitary ensemble defined by the perturbed chebyshev weight w x 1 x2 12 tn2 x2 x 1 1 tn 1 the algebraic singularity coalesces with the end point of the support of the weight function namely tn 1 the hard edge as n we use the riemann hilbert approach the method of deift and zhou to obtain the asymptotics of the polynomials orthogonal with respect to w x we show that the universality property is preserved the asymptotic behavior of the eigenvalue correlations in the bulk of the spectrum is described in terms of the sine kernel the main result is on the local behavior at the edge of the spectrum the limit kernel at the edge when tn 1 o n 2 is described by the function for a specific solution of the painleve v equation we also show that when tn varies to a fixed t 1 the painleve v kernel transits to the bessel kernel j 12 on the other hand when tn approaches 1 the painleve v kernel can be approximated by another bessel kernel j 12

generalization of the h deformation to higher dimensions we construct glh 3 from glq 3 by a singular map we show that there exist two singular maps which map glq 3 to new quantum groups we also construct their r matrices and will show that although the maps are singular their r matrices are not then we generalize these singular maps to the case gl n and for cn series

spatially varying coefficient model for neuroimaging data with jump discontinuities motivated by recent work on studying massive imaging data in various neuroimaging studies we propose a novel spatially varying coefficient model svcm to capture the varying association between imaging measures in a three dimensional volume or two dimensional surface with a set of covariates two stylized features of neuorimaging data are the presence of multiple piecewise smooth regions with unknown edges and jumps and substantial spatial correlations to specifically account for these two features svcm includes a measurement model with multiple varying coefficient functions a jumping surface model for each varying coefficient function and a functional principal component model we develop a three stage estimation procedure to simultaneously estimate the varying coefficient functions and the spatial correlations the estimation procedure includes a fast multiscale adaptive estimation and testing procedure to independently estimate each varying coefficient function while preserving its edges among

universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles this paper is devoted to the rigorous proof of the universality conjecture of random matrix theory according to which the limiting eigenvalue statistics ofn n random matrices within spectral intervals ofo n 1 is determined by the type of matrix real symmetric hermitian or quaternion real and by the density of states we prove this conjecture for a certain class of the hermitian matrix ensembles that arise in the quantum field theory and have the unitary invariant distribution defined by a certain function the potential in the quantum field theory satisfying some regularity conditions

exact solution and surface critical behaviour of open cyclic sos lattice models we consider the l state cyclic solid on solid lattice models under a class of open boundary conditions the integrable boundary face weights are obtained by solving the reflection equations functional relations for the fused transfer matrices are presented for both periodic and open boundary conditions the eigenspectra of the unfused transfer matrix is obtained from the functional relations using the analytic bethe ansatz for a special case of crossing parameter the finite size corrections to the eigenspectra of the critical models are obtained from which the corresponding conformal dimensions follow the calculation of the surface free energy away from criticality yields two surface specific heat exponents and where coprime to l these results are in agreement with the scaling relations and

tracy widom distributions in critical unitary random matrix ensembles and the coupled painleve ii system we study fredholm determinants of the painleve ii and painleve xxxiv kernels in certain critical unitary random matrix ensembles these determinants describe special gap probabilities of eigenvalues we obtain tracy widom formulas for the fredholm determinants which are explicitly given in terms of integrals involving a family of distinguished solutions to the coupled painleve ii system in dimension four moreover the large gap asymptotics for these fredholm determinants are derived where the constant terms are given explicitly in terms of the riemann zeta function

high dimensional lattice gases we investigate the critical behaviour of hard core lattice gases in four five and six dimensions by means of monte carlo simulations in order to suppress critical slowing down we use a geometrical cluster monte carlo algorithm in particular nearest neighbour exclusion lattice gases on simple hypercubic lattices are investigated these models undergo ising like ordering transitions where the majority of the lattice gas particles settle on one of two sublattices a finite size scaling analysis of the simulation data confirms that these lattice gases display classical critical behaviour the results agree with the renormalization predictions at and above the upper critical dimensionality in particular the predicted value of the binder cumulant is confirmed

universality of a double scaling limit near singular edge points in random matrix models we consider unitary random matrix ensembles z n s t 1 e n tr v s t m dm on the space of hermitian n n matrices m where the confining potential vs t is such that the limiting mean density of eigenvalues as n and s t 0 vanishes like a power 5 2 at a singular endpoint of its support the main purpose of this paper is to prove universality of the eigenvalue correlation kernel in a double scaling limit the limiting kernel is built out of functions associated with a special solution of the pi2 equation which is a fourth order analogue of the painleve i equation in order to prove our result we use the well known connection between the eigenvalue correlation kernel and the riemann hilbert rh problem for orthogonal polynomials together with the deift zhou steepest descent method to analyze the rh problem asymptotically the key step in the asymptotic analysis will be the construction of a parametrix near the singular endpoint for which we use the model rh problem for the special solution of the pi2 equation

on eigenvalues of the sum of two random projections we study the behavior of eigenvalues of matrix p n q n where p n and q n are two n by n random orthogonal projections we relate the joint eigenvalue distribution of this matrix to the jacobi matrix ensemble and establish the universal behavior of eigenvalues for large n the limiting local behavior of eigenvalues is governed by the sine kernel in the bulk and by either the bessel or the airy kernel at the edge depending on parameters we also study an exceptional case when the local behavior of eigenvalues of p n q n is not universal in the usual sense

spectral statistics of erdős renyi graphs ii eigenvalue spacing and the extreme eigenvalues we consider the ensemble of adjacency matrices of erd s renyi random graphs i e graphs on n vertices where every edge is chosen independently and with probability p p n we rescale the matrix so that its bulk eigenvalues are of order one under the assumption p n gg n 2 3 we prove the universality of eigenvalue distributions both in the bulk and at the edge of the spectrum more precisely we prove 1 that the eigenvalue spacing of the erd s renyi graph in the bulk of the spectrum has the same distribution as that of the gaussian orthogonal ensemble and 2 that the second largest eigenvalue of the erd s renyi graph has the same distribution as the largest eigenvalue of the gaussian orthogonal ensemble as an application of our method we prove the bulk universality of generalized wigner matrices under the assumption that the matrix entries have at least 4 e moments

semiclassical asymptotics of orthogonal polynomials riemann hilbert problem and universality in the matrix model we derive semiclassical asymptotics for the orthogonal polynomials pn z on the line with respect to the exponential weight exp inv z where v z is a double well quartic polynomial in the limit when n n 1 we assume that n n cri for some 0 where cr is the critical value which separates orthogonal polynomials with two cuts from the ones with one cut simultaneously we derive semiclassical asymptotics for the recursive coe cients of the orthogonal polynomials and we show that these coe cients form a cycle of period two which drifts slowly with the change of the ratio n n the proof of the semiclassical asymptotics is based on the methods of the theory of integrable systems and on the analysis of the appropriate matrix riemannhilbert problem as an application of the semiclassical asymptotics of the orthogonal polynomials we prove the universality of the local distribution of eigenvalues in the matrix model with the double well quartic interaction in the presence of two cuts contents

diffusion tensor imaging structural adaptive smoothing abstract diffusion tensor imaging dti data is characterized by a high noise level thus estimation errors of quantities like anisotropy indices or the main diffusion direction used for fiber tracking are relatively large and may significantly confound the accuracy of dti in clinical or neuroscience applications besides pulse sequence optimization noise reduction by smoothing the data can be pursued as a complementary approach to increase the accuracy of dti here we suggest an anisotropic structural adaptive smoothing procedure which is based on the propagation separation method and preserves the structures seen in dti and their different sizes and shapes it is applied to artificial phantom data and a brain scan we show that this method significantly improves the quality of the estimate of the diffusion tensor by means of both bias and variance reduction and hence enables one either to reduce the number of scans or to enhance the input for subsequent analysis such as fiber tracking

uniform asymptotics of the pollaczek polynomials via the riemann hilbert approach the pollaczek weight is an example of the non szeg class in this paper we investigate the asymptotics of the pollaczek polynomials via the riemannhilbert approach in the analysis the original e

asymptotics of recurrence coefficients for orthonormal polynomials on the line magnus s method revisited we use freud equations to obtain the main term in the asymptotic expansion of the recurrence coefficients associated with orthonormal polynomials pn w2 for weights w w exp q on the real line where q is an even polynomial of fixed degree with nonnegative coefficients or where q x exp x2m m 1 here w x x for some real 1

norm dependent random matrix ensembles in external field and supersymmetry the class of norm dependent random matrix ensembles is studied in the presence of an external field the probability density of the ensemble depends on the trace of the squared random matrices but is otherwise arbitrary an exact mapping to superspace is performed a transformation formula is derived which gives the probability density in superspace as a single integral over the probability density in ordinary space this is done for orthogonal unitary and symplectic symmetries in the case of unitary symmetry some explicit results for the correlation functions are derived

exact enumeration results for self avoiding walks on the honeycomb lattice attached to a surface we consider self avoiding walks on the honeycomb lattice interacting with a surface with different energies associated between sites in contact with a linear boundary to the left of the origin and those in contact with the right of the boundary we numerically confirm recent exact results for the polymer adsorption transition and corresponding critical exponents with mixed ordinary and special boundary conditions the phase diagram is elucidated with the aid of some rigorous arguments

limit correlation functions for fixed trace random matrix ensembles universal limits for the eigenvalue correlation functions in the bulk of the spectrum are shown for a class of nondeterminantal random matrices known as the fixed trace or the hilbert schmidt ensemble these universal limits have been proved before for determinantal hermitian matrix ensembles and for some special classes of the wigner random matrices

on asymptotic expansions and scales of spectral universality in band random matrix ensembles we consider real random symmetric n n matrices h of the band type form with characteristic length b the matrix entries are independent gaussian random variables and have the variance proportional to where u t vanishes at infinity we study the resolvent in the limit and obtain the explicit expression for the leading term of the first correlation function of the normalized trace

compact support probability distributions in random matrix theory we consider a generalization of the fixed and bounded trace ensembles introduced by bronk and rosenzweig up to an arbitrary polynomial potential in the large n limit we prove that the two are equivalent and that their eigenvalue distribution coincides with that of the canonical ensemble with measure exp ntr v m the mapping of the corresponding phase boundaries is illuminated in an explicit example in the case of a gaussian potential we are able to derive exact expressions for the one and two point correlator for finite n having finite support

the nonlinear steepest descent approach to the singular asymptotics of the second painleve transcendent abstract we consider the real valued solutions of the second painleve equation on the real line the two parameter family of the solutions having singular asymptotics as x or and x is studied with the help of the deift zhou nonlinear steepest descent method explicit evaluation in terms of trigonometric functions of the singular leading orders of the asymptotics is carried out and the corresponding connection formulae obtained a novel methodological feature is the appearance of the soliton type riemann hilbert problem in the course of the implementation of the deift zhou scheme for the riemann hilbert problem corresponding to the second painleve equation within the riemann hilbert isomonodromy approach the result of the paper reproduces previously known formulae derived by kapaev via the original isomonodromy technique

nonuniversality of compact support probability distributions in random matrix theory the two point resolvent is calculated in the large n limit for thegeneralized fixed and bounded trace ensembles it is shown to disagree with theone of the canonical gaussian ensemble by a non universal part which is givenexplicitly for all monomial potentials v m m 2p moreover we prove thatfor the generalized fixed and bounded trace ensemble all k point resolventsagree in the large n limit despite their non universality

universality of the local spacing distribution in certain ensembles of hermitian wigner matrices consider an n n hermitian random matrix with independent entries not necessarily gaussian a so called wigner matrix it has been conjectured that the local spacing distribution i e the distribution of the distance between nearest neighbour eigenvalues in some part of the spectrum is in the limit as n the same as that of hermitian random matrices from gue we prove this conjecture for a certain subclass of hermitian wigner matrices