central limit theorem for linear eigenvalue statistics of random matrices with independent entries we consider n x n real symmetric and hermitian wigner random matrices n 1 2 w with independent modulo symmetry condition entries and the null sample covariance matrices n 1 x x with independent entries of m x n matrix x assuming first that the 4th cumulant excess 4 of entries of w and x is zero and that their 4th moments satisfy a lindeberg type condition we prove that linear statistics of eigenvalues of the above matrices satisfy the central limit theorem clt as n m m n c 0 oo with the same variance as for gaussian matrices if the test functions of statistics are smooth enough essentially of the class c 5 this is done by using a simple interpolation trick from the known results for the gaussian matrices and the integration by parts presented in the form of certain differentiation formulas then by using a more elaborated version of the techniques we prove the clt in the case of nonzero excess of entries again for essentially 5 test function here the variance of statistics contains an additional term proportional to 4 the proofs of all limit theorems follow essentially the same scheme

bulk universality for generalized wigner matrices consider n n hermitian or symmetric random matrices h where the distribution of the i j matrix element is given by a probability measure ij with a subexponential decay let sigma ij 2 be the variance for the probability measure ij with the normalization property that sum i sigma 2 ij 1 for all j under essentially the only condition that c le n sigma ij 2 le c 1 for some constant c 0 we prove that in the limit n the eigenvalue spacing statistics of h in the bulk of the spectrum coincide with those of the gaussian unitary or orthogonal ensemble gue or goe we also show that for band matrices with bandwidth m the local semicircle law holds to the energy scale m 1

mean field upper and lower bounds on lyapunov exponents we extend the supersymmetric field theory formulation of random walks in random po tentials developed in earlier papers we obtain the mean field rate of decay for the squared green s function at the critical energy this requires studying supersymmetric field theories which are quar tic in the grassmann variables we also obtain both the mean field upper and lower bounds at the critical energy for lyapunov exponents for unbounded potentials the upper bound part extends the corresponding result in an earlier work

on the second mixed moment of the characteristic polynomials of 1d band matrices we consider the asymptotic behavior of the second mixed moment of the characteristic polynomials of 1d gaussian band matrices i e of the hermitian n n matrices h n with independent gaussian entries such that h ij h lk ik jl j ij where j w 2 triangle 1 1 assuming that w 2 n 1 theta 0 theta leq 1 we show that the moment s asymptotic behavior as n to infty in the bulk of the spectrum coincides with that for the gaussian unitary ensemble

augmented is lm model based on particular functions abstract in this paper we create our own static and dynamic augmented is lm model based on concrete particular functions these functions represent a mathematical and analytical description of economical quantities on which the is lm model is based and whose properties follow from the corresponding economical theories we prove several theorems about parameters of this model we describe dynamical behaviour of this system and prove un stability of the singular points including their classification then we demonstrate this model on some illustrative examples and we suggest our own econometric non linear regression models of the particular functions courses

local kesten mckay law for random regular graphs we study the adjacency matrices of random d regular graphs with large but fixed degree d in the bulk of the spectrum 2 sqrt d 1 varepsilon 2 sqrt d 1 varepsilon down to the optimal spectral scale we prove that the green s functions can be approximated by those of certain infinite tree like few cycles graphs that depend only on the local structure of the original graphs this result implies that the kesten mckay law holds for the spectral density down to the smallest scale and the complete delocalization of bulk eigenvectors our method is based on estimating the green s function of the adjacency matrices and a resampling of the boundary edges of large balls in the graphs

subordination for the sum of two random matrices this paper is about the relation of random matrix theory and the subordination phenomenon in complex analysis we find that the resolvent of the sum of two random matrices is approximately subordinated to the resolvents of the original matrices we estimate the error terms in this relation and in the subordination relation for the traces of the resolvents this allows us to prove a local limit law for eigenvalues and a delocalization result for eigenvectors of the sum of two random matrices in addition we use subordination to determine the limit of the largest eigenvalue for the rank one deformations of unitary invariant random matrices

semiparametric analysis of random effects linear models from binary panel data andersen 1970 considered the problem of inference on random effects linear models from binary response panel data he showed that inference is possible if the disturbances for each panel member are known to be white noise with the logistic distribution and if the observed explanatory variables vary over time a conditional maximum likelihood estimator consistently estimates the model parameters up to scale the present paper shows that inference remains possible if the disturbances for each panel member are known only to be time stationary with unbounded support and if the explanatory variables vary enough over time a conditional version of the maximum score estimator manski 1975 1985 consistently estimates the model parameters up to scale

the eigenvector moment flow and local quantum unique ergodicity we prove that the distribution of eigenvectors of generalized wigner matrices is universal both in the bulk and at the edge this includes a probabilistic version of local quantum unique ergodicity and asymptotic normality of the eigenvector entries the proof relies on analyzing the eigenvector flow under the dyson brownian motion the key new ideas are 1 the introduction of the eigenvector moment flow a multi particle random walk in a random environment 2 an effective estimate on the regularity of this flow based on maximum principle and 3 optimal finite speed of propagation holds for the eigenvector moment flow with very high probability

statistical analysis of chiral structured ensembles role of matrix constraints we analyze the statistical properties of complex systems with specific conservation laws and symmetry conditions which lead to various constraints and thereby structures in their matrix representation an increase of constraints leads to a variation of the spectral statistics from wigner dyson to poisson limits but the eigenfunction statistics remains weakly multifractal in the bulk for some constraints the statistics not only lies between the two limits but is size independent too thus indicating a critical point our results also reveal an important trend while the spectral statistics is strongly sensitive to the number of independent matrix elements the eigenfunction statistics seems to be affected mainly by their relative strengths this is contrary to a previously held belief of a one to one relation between the statistics of the eigenfunctions and eigenvalues e g associating poisson statistics to the localized eigenfunctions and wigner dyson statistics to delocalized ones this also indicates the existence of new universality classes based on the matrix constraints different from the 10 already known symmetry based classes

brownian motions of ellipsoids on donne un traitement elementaire des mouvements browniens d ellipsoides et en particulier des resultats remarquables obtenus par dynkin

a dynamic is lm business cycle model with two time delays in capital accumulation equation in this paper we analyze a augmented is lm business cycle model with the capital accumulation equation that two time delays are considered in investment processes according to kalecki s idea applying stability switch criteria and hopf bifurcation theory we prove that time delays cause the equilibrium to lose or gain stability and hopf bifurcation occurs

spontaneous symmetry breaking of a hyperbolic sigma model in three dimensions non linear sigma models that arise from the supersymmetric approach to disordered electron systems contain a non compact bosonic sector we study the model with target space h2 the two hyperboloid with isometry group su 1 1 and prove that in three dimensions moments of the fields are finite in the thermodynamic limit thus the non compact symmetry su 1 1 is spontaneously broken the bound on moments is compatible with the presence of extended states

large complex correlated wishart matrices fluctuations and asymptotic independence at the edges we study the asymptotic behavior of eigenvalues of large complex correlated wishart matrices at the edges of the limiting spectrum in this setting the support of the limiting eigenvalue distribution may have several connected components under mild conditions for the population matrices we show that for every generic positive edge of that support there exists an extremal eigenvalue which converges almost surely towards that edge and fluctuates according to the tracy widom law at the scale n 2 3 moreover given several generic positive edges we establish that the associated extremal eigenvalue fluctuations are asymptotically independent finally when the leftmost edge is the origin we prove that the smallest eigenvalue fluctuates according to the hard edge tracy widom law at the scale n 2 as an application an asymptotic study of the condition number of large correlated wishart matrices is provided

semiparametric estimation of a work trip mode choice model abstract a binary response model of choice between automobile and transit for travel to work is estimated using fixed and random coefficients probit specifications a semiparametric single index specification and a semiparametric specification that permits arbitrary heteroskedasticity of unknown form the estimation methods are maximum likelihood for the probit models semiparametric quasi maximum likelihood for the single index model and maximum score and smoothed maximum score for the arbitrarily heteroskedastic model specification tests result in rejection of the fixed coefficients probit and single index models but not of the random coefficients probit model and the semiparametric model with arbitrary heteroskedasticity

eigenvector localization for random band matrices with power law band width it is shown that certain ensembles of random matrices with entries that vanish outside a band around the diagonal satisfy a localization condition on the resolvent which guarantees that eigenvectors have strong overlap with a vanishing fraction of standard basis vectors provided the band width w raised to a power remains smaller than the matrix size n for a gaussian band ensemble with matrix elements given by i i d centered gaussians within a band of width w the estimate 8 holds

overlaps between eigenvectors of correlated random matrices we obtain general exact formulas for the overlaps between the eigenvectors of large correlated random matrices with additive or multiplicative noise these results have potential applications in many different contexts from quantum thermalization to high dimensional statistics we find that the overlaps only depend on measurable quantities and do not require the knowledge of the underlying true noiseless matrices we apply our results to the case of empirical correlation matrices that allow us to estimate reliably the width of the spectrum of the true correlation matrix even when the latter is very close to the identity we illustrate our results on the example of stock returns correlations which clearly reveal a nontrivial structure for the bulk eigenvalues we also apply our results to the problem of matrix denoising in high dimensions

distribution free maximum likelihood estimator of the binary choice model1 is a given function of the exogenous variables z and unknown parameters 9 representing the systematic component of the utility difference and f is the distribution function of the random component of the utility difference this paper describes a method of estimating the parameters 9 without assuming any functional form for the distribution function f and proves that this estimator is consistent f is also consistently estimated the method uses maximum likelihood estimation in which the likelihood is maximized not only over the parameter 9 but also over a space which contains all distribution functions

exact computation of max weighted score estimators abstract we show that exact computation of a family of max weighted score estimators including manski s max score estimator can be achieved efficiently by reformulating them as mixed integer programs mip with disjunctive constraints the advantage of our mip formulation is that estimates are exact and can be computed using widely available solvers in reasonable time in a classic work trip mode choice application our method delivers exact estimates that lead to a different economic interpretation of the data than previous heuristic estimates in a small monte carlo study we find that our approach is computationally efficient for usual estimation problem sizes

random matrices tail bounds for gaps between eigenvalues gaps or spacings between consecutive eigenvalues are a central topic in random matrix theory the goal of this paper is to study the tail distribution of these gaps in various random matrix models we give the first repulsion bound for random matrices with discrete entries and the first super polynomial bound on the probability that a random graph has simple spectrum along with several applications

invariant measures and arithmetic quantum unique ergodicity we classify measures on the locally homogeneous space i sl 2 r l which are invariant and have positive entropy under the diagonal subgroup of sl 2 r and recurrent under l this classification can be used to show arithmetic quantum unique ergodicity for compact arithmetic surfaces and a similar but slightly weaker result for the finite volume case other applications are also presented in the appendix joint with d rudolph we present a maximal ergodic theorem related to a theorem of hurewicz which is used in theproof of the main result

edge universality of beta ensembles we prove the edge universality of the beta ensembles for any beta ge 1 provided that the limiting spectrum is supported on a single interval and the external potential is fancyscript c 4 and regular we also prove that the edge universality holds for generalized wigner matrices for all symmetry classes moreover our results allow us to extend bulk universality for beta ensembles from analytic potentials to potentials in class fancyscript c 4

the distribution of overlaps between eigenvectors of ginibre matrices we study the overlaps between eigenvectors of nonnormal matrices they quantify the stability of the spectrum and characterize the joint eigenvalues increments under dyson type dynamics well known work by chalker and mehlig calculated the expectation of these overlaps for complex ginibre matrices for the same model we extend their results by deriving the distribution of diagonal overlaps the condition numbers and their correlations we prove i convergence of condition numbers for bulk eigenvalues to an inverse gamma distribution more generally we decompose the quenched overlap i e conditioned on eigenvalues as a product of independent random variables ii asymptotic expectation of off diagonal overlaps both for microscopic or mesoscopic separation of the corresponding eigenvalues iii decorrelation of condition numbers associated to eigenvalues at mesoscopic distance at polynomial speed in the dimension iv second moment asymptotics to identify the fluctuations order for off diagonal overlaps when the related eigenvalues are separated by any mesoscopic scale v a new formula for the correlation between overlaps for eigenvalues at microscopic distance both diagonal and off diagonal these results imply estimates on the extreme condition numbers the volume of the pseudospectrum and the diffusive evolution of eigenvalues under dyson type dynamics at equilibrium

delocalization for a class of random block band matrices we consider n times n hermitian random matrices h consisting of blocks of size m ge n 6 7 the matrix elements are i i d within the blocks close to a gaussian in the four moment matching sense but their distribution varies from block to block to form a block band structure with an essential band width m we show that the entries of the green s function g z h z 1 satisfy the local semicircle law with spectral parameter z e mathbf i eta down to the real axis for any eta gg n 1 using a combination of the supersymmetry method inspired by shcherbina j stat phys 155 3 466 499 2014 and the green s function comparison strategy previous estimates were valid only for eta gg m 1 the new estimate also implies that the eigenvectors in the middle of the spectrum are fully delocalized

extended states in the anderson model on the bethe lattice abstract we prove that the anderson hamiltonianh von the bethe lattice has extended states for small disorder more precisely given any closed intervalicontained in the interior of the spectrum of the laplacian on the bethe lattice we prove that for small disorderh has purely absolutely continuous spectrum iniwith probability one i e ac h i iand pp h i sc h i with probability one and its integrated density of states is continuously differentiable on the intervali

no eigenvalues outside the support of the limiting spectral distribution of large dimensional sample covariance matrices let b n 1 n t n 1 2 x n x n t n 1 2 where x n is n times n with i i d complex standardized entries having finite fourth moment and t n 1 2 is a hermitian square root of the nonnegative definite hermitian matrix t n it is known that as n to infty if n n converges to a positive number and the empirical distribution of the eigenvalues of t n converges to a proper probability distribution then the empirical distribution of the eigenvalues of b n converges a s to a nonrandom limit in this paper we prove that under certain conditions on the eigenvalues of t n for any closed interval outside the support of the limit with probability 1 there will be no eigenvalues in this interval for all n sufficiently large

superbosonization formula and its application to random matrix theory starting from gaussian random matrix models we derive a new supermatrix field theory model in contrast to the conventional non linear sigma models the new model is applicable for any range of correlations of the elements of the random matrices we clarify the domain of integration for the supermatrices and give a demonstration of how the model works by calculating the density of states for an ensemble of almost diagonal matrices it is also shown how one can reduce the supermatrix model to the conventional sigma model

identification of binary response models abstract this article studies identification of the threshold crossing model of binary response most research on binary response has considered specific estimators and tests the study of identification exposes the foundations of binary response analysis by making explicit the assumptions needed to justify different methods it also clarifies the connections between reduced form and structural analyses of binary response data assume that the binary outcome z is determined by an observable random vector x and by an unobservable scalar u through a model z 1 x u 0 also assume that fu x the probability distribution of u conditional on x is continuous and strictly increasing given these maintained assumptions we investigate the identifiability of given the following restrictions on the distributions fu x x x mean independence quantile independence index sufficiency statistical independence and statistical independence with the distribution known we find that mean independence has n

the isotropic semicircle law and deformation of wigner matrices we analyze the spectrum of additive finite rank deformations of n n wigner matrices h the spectrum of the deformed matrix undergoes a transition associated with the creation or annihilation of an outlier when an eigenvalue di of the deformation crosses a critical value 1 this transition happens on the scale we allow the eigenvalues di of the deformation to depend on n under the condition we make no assumptions on the eigenvectors of the deformation in the limit n we identify the law of the outliers and prove that the nonoutliers close to the spectral edge have a universal distribution coinciding with that of the extremal eigenvalues of a gaussian matrix ensemble a key ingredient in our proof is the isotropic local semicircle law which establishes optimal high probability bounds on where m z is the stieltjes transform of wigner s semicircle law and v w are arbitrary deterministic vectors 2013 wiley periodicals inc

absence of diffusion in the anderson tight binding model for large disorder or low energy we prove that the green s function of the anderson tight binding hamiltonian decays exponentially fast at long distances on v with probability 1 we must assume that either the disorder is large or the energy is sufficiently low our proof is based on perturbation theory about an infinite sequence of block hamiltonians and is related to kam methods

bulk universality for generalized wigner matrices with few moments in this paper we consider n times n real generalized wigner matrices whose entries are only assumed to have finite 2 varepsilon th moment for some fixed but arbitrarily small varepsilon 0 we show that the stieltjes transforms m n z of these matrices satisfy a weak local semicircle law on the nearly smallest possible scale when eta mathfrak i z is almost of order n 1 as a consequence we establish bulk universality for local spectral statistics of these matrices at fixed energy levels both in terms of eigenvalue gap distributions and correlation functions meaning that these statistics converge to those of the gaussian orthogonal ensemble in the large n limit

local semicircle law for random regular graphs we consider random d regular graphs on n vertices with degree d at least log n 4 we prove that the green s function of the adjacency matrix and the stieltjes transform of its empirical spectral measure are well approximated by wigner s semicircle law down to the optimal scale given by the typical eigenvalue spacing up to a logarithmic correction aside from well known consequences for the local eigenvalue distribution this result implies the complete isotropic delocalization of all eigenvectors and a probabilistic version of quantum unique ergodicity 2017 wiley periodicals inc

optimal bandwidth selection in nonparametric regression function estimation kernel estimators of an unknown multivariate regression function are investigated a bandwidth selection rule is considered which can be formulated in terms of cross validation under mild assumptions on the kernel and the unknown regression function it is seen that this rule is asymptotically optimal

superbosonization of invariant random matrix ensembles superbosonization is a new variant of the method of commuting and anti commuting variables as used in studying random matrix models of disordered and chaotic quantum systems we here give a concise mathematical exposition of the key formulas of superbosonization conceived by analogy with the bosonization technique for dirac fermions the new method differs from the traditional one in that the superbosonization field is dual to the usual hubbard stratonovich field the present paper addresses invariant random matrix ensembles with symmetry group un on or uspn giving precise definitions and conditions of validity in each case the method is illustrated at the example of wegner s n orbital model superbosonization promises to become a powerful tool for investigating the universality of spectral correlation functions for a broad class of random matrix ensembles of non gaussian and or non invariant type

localization at large disorder and at extreme energies an elementary derivation the work presents a short proof of localization under the conditions of either strong disorder 0 or extreme energies for a wide class of self adjoint operators with random matrix elements acting inl 2 spaces a prototypical example is the discrete schrodinger operatorh u 0 x v x onz d d 1 withu 0 x a specified background potential and v x generated as random variables the general results apply to operators with replaced by a non local self adjoint operatort whose matrix elements satisfy y t x y s const uniformly inx for somes 1 localization means here that within a specified energy range the spectrum ofh is of the pure point type or equivalently the wave functions do not spread indefinitely under the unitary time evolution generated byh the effect is produced by strong disorder in either the potential or in the off diagonal matrix elementst x y under rapid decay oft x y the corresponding eigenfunctions are also proven to decay exponentially the method is based on resolvent techniques the central technical ideas include the use of low moments of the resolvent kernel i e g e x y s withs small enough 1 to avoid the divergence caused by the distribution s cauchy tails and an effective use of the simple form of the dependence ofg e x y on the individual matrix elements ofh in elucidating the implications of the fundamental equation h e g e x x 0 x x0 this approach simplifies previous derivations of localization results avoiding the small denominator difficulties which have been hitherto encountered in the subject it also yields some new results which include localization under the following sets of conditions i potentials with an inhomogeneous non random partu 0 x ii the bethe lattice iii operators with very slow decay in the off diagonal terms t x y 1 x y d e and iv localization produced by disordered boundary conditions

fixed energy universality of dyson brownian motion abstract we consider dyson brownian motion for classical values of with deterministic initial data v we prove that the local eigenvalue statistics coincide with the goe gue in the fixed energy sense after time t 1 n if the density of states of v is bounded above and below down to scales t in a window of size l t our results imply that fixed energy universality holds for essentially any random matrix ensemble for which averaged energy universality was previously known our methodology builds on the homogenization theory developed in 16 which reduces the microscopic problem to a mesoscopic problem as an auxiliary result we prove a mesoscopic central limit theorem for linear statistics of various classes of test functions for classical dyson brownian motion

approximation to stable law by the lindeberg principle abstract by the lindeberg principle we develop in this paper an approximation to one dimensional possibly asymmetric stable distributions with 0 2 in the smooth wasserstein distance it is the first time that the general stable central limit theorem is proved by the lindeberg principle and that this theorem with 0 1 is proved by a new method other than fourier analysis our main tools are a taylor like expansion and a kolmogorov forward equation

spectral statistics of erdős renyi graphs i local semicircle law we consider the ensemble of adjacency matrices of erd s renyi random graphs that is 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 we prove that as long as pn with a speed at least logarithmic in n the density of eigenvalues of the erd s renyi ensemble is given by the wigner semicircle law for spectral windows of length larger than n 1 up to logarithmic corrections as a consequence all eigenvectors are proved to be completely delocalized in the sense that the l norms of the l2 normalized eigenvectors are at most of order n 1 2 with a very high probability the estimates in this paper will be used in the companion paper spectral statistics of erd s renyi graphs ii eigenvalue spacing and the extreme eigenvalues 2011 preprint to prove the universality of eigenvalue distributions both in the bulk and at the spectral edges under the further restriction that pn n2 3

a semiparametric structural analysis of the idling of cement kilns abstract this paper estimates the decision rule for idling capital at the micro firm level this is done for a cement kiln by using data from the us cement industry the decision rule for not operating a kiln at a given point in time is the standard rule do not operate if the average variable cost exceeds price the parameters of this discrete decision rule are semiparametrically estimated using the maximum score method and the estimates are then compared with the logit maximum likelihood estimates in terms of their values as well as predictions

operational characteristics of maximum score estimation abstract this paper reports on the operational characteristics of maximum score estimation of a linear model from binary response data a series of previous articles have shown that in theory the maximum score method makes possible binary response analysis under very weak distributional assumptions here we present evidence on the properties of maximum score estimation in practice after reviewing the known asymptotic theory of maximum score estimation the paper describes an algorithm for maximum score estimation and characterizes its performance then findings from a monte carlo study comparing maximum score and logit maximum likelihood estimation are reported finally the accuracy of bootstrap estimation of maximum score root mean square errors is evaluated

an estimate for the average spectral measure of random band matrices for a class of random band matrices of band width w we prove regularity of the average spectral measure at scales w 0 99 and find its asymptotics at these scales

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

from gumbel to tracy widom the tracy widom distribution that has been much studied in recent years can be thought of as an extreme value distribution we discuss interpolation between the classical extreme value distribution exp exp x the gumbel distribution and the tracy widom distribution there is a family of determinantal processes whose edge behaviour interpolates between a poisson process with density exp x and the airy kernel point process this process can be obtained as a scaling limit of a grand canonical version of a random matrix model introduced by moshe neuberger and shapiro we also consider the deformed gue ensemble m m 0 sqrt 2s v with m 0 diagonal with independent elements and v from gue here we do not see a transition from tracy widom to gumbel but rather a transition from tracy widom to gaussian

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

entropy and the localization of eigenfunctions we study the large eigenvalue limit for the eigenfunctions of the laplacian on a compact manifold of negative curvature in fact we only assume that the geodesic flow has the anosov property in the semi classical limit we prove that the wigner measures associated to eigenfunctions have positive metric entropy in particular they cannot concentrate entirely on closed geodesies 1 introduction statement of results we consider a compact riemannian manifold m of dimension d 2 and assume that the geodesic flow ter acting on the unit tangent bundle of m has a chaotic behaviour this refers to the asymptotic properties of the flow when time t tends to infinity ergodicity mixing hyperbolicity we assume here that the geodesic flow has the anosov property the main example being the case of negatively curved manifolds the words quantum chaos express the intuitive idea that the chaotic features of the geodesic flow should imply certain special features for the corresponding quantum dynamical system that is according to schrodinger the unitary flow exp iht t r acting on the hilbert space l2 m where a stands for the laplacian on m and h is proportional to the planck constant recall that the quantum flow converges in a sense to the classical flow g in the so called semi classical limit h 0 one can imagine that for small values ofh the quantum system will inherit certain qualitative properties of the classical flow one expects for instance a very different behaviour of eigenfunctions of the laplacian or the distribution of its eigenvalues if the geodesic flow is anosov or in the other extreme completely integrable see sa95 the convergence of the quantum flow to the classical flow is stated in the egorov theorem consider one of the usual quantization procedures op which associates an operator oph a acting on l2 m to every smooth compactly supported function a e c t m on the cotangent bundle t m according to the egorov theorem we have for any fixed t

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

asymptotic theory of eigenvectors for random matrices with diverging spikes characterizing the asymptotic distributions of eigenvectors for large random matrices poses important challenges yet can provide useful insights into a range of statistical applications to this en

semiparametric analysis of discrete response asymptotic properties of the maximum score estimator abstract this paper is concerned with the estimation of the model med y x x from a random sample of observations on sgn y x manski 1975 introduced the maximum score estimator of the normalized parameter vector in the present paper strong consistency is proved it is also proved that the maximum score estimate lies outside any fixed neighborhood of with probability that goes to zero at exponential rate

existence of chaos in the plane r 2 and its application in macroeconomics the existence of devaney li yorke and distributional chaos in the plane r 2 the chaos in a continuous dynamical system generated by euler equation branching all combinations of hyperbolic singular points of both branches admit chaos the chaotic set of solutions associated with chaotic subset of r 2 exists new overall macroeconomic equilibrium model called is lm qy ml model the existence of chaos in an economy the devaney li yorke and distributional chaos in the plane r 2 can occur in the continuous dynamical system generated by euler equation branching euler equation branching is a type of differential inclusion x f x g x where f g x r n r n are continuous and f x g x in every point x x stockman and raines 2010 defined the so called chaotic set in the plane r 2 whose existence leads to the existence of devaney li yorke and distributional chaos in this paper we follow up on stockman and raines 2010 and we show that chaos in the plane r 2 is always admitted for hyperbolic singular points in both branches not lying in the same point in r 2 but the chaos existence is also caused by a set of solutions of euler equation branching we research this set of solutions in the second part we create the new overall macroeconomic equilibrium model called is lm qy ml model this model is based on the fundamental macroeconomic equilibrium model called is lm model we model an inflation effect an endogenous money supply an economic cycle etc in addition to the original is lm model we research the dynamical behaviour of this new is lm qy ml model and show when a chaos exists with relevant economic interpretation

a generalization of the lindeberg principle we generalize lindeberg s proof of the central limit theorem to an invariance principle for arbitrary smooth functions of independent and weakly dependent random variables the result is applied to get a similar theorem for smooth functions of exchangeable random variables this theorem allows us to identify for the first time the limiting spectral distributions of wigner matrices with exchangeable entries

sieving for mass equidistribution we approach the holomorphic analogue to the quantum unique ergodicity conjecture through an application of the large sieve we deal with shifted convolution sums as in ho109 with various simplifications in our analysis due to the knowledge of the ramanujan petersson conjecture in this holomorphic case

exact separation of eigenvalues of large dimensional sample covariance matrices let b n 1 n t n 1 2 x n x n t n 1 2 where x n is n times n with i i d complex standardized entries having finite fourth moment and t n 1 2 is a hermitian square root of the nonnegative definite hermitian matrix t n it was shown in an earlier paper by the authors that under certain conditions on the eigenvalues of t n with probability 1 no eigenvalues lie in any interval which is outside the support of the limiting empirical distribution known to exist for all large n for these n the interval corresponds to one that separates the eigenvalues of t n the aim of the present paper is to prove exact separation of eigenvalues that is with probability 1 the number of eigenvalues of b n and t n lying on one side of their respective intervals are identical for all large n

on delocalization of eigenvectors of random non hermitian matrices we study delocalization of null vectors and eigenvectors of random matrices with i i d entries let a be an n times n random matrix with i i d real subgaussian entries of zero mean and unit variance we show that with probability at least 1 e log 2 n begin aligned min limits i subset n i m vert mathbf v i vert ge frac m 3 2 n 3 2 log cn vert mathbf v vert end aligned for any real eigenvector mathbf v and any m in log c n n where mathbf v i denotes the restriction of mathbf v to i further when the entries of a are complex with i i d real and imaginary parts we show that with probability at least 1 e log 2 n all eigenvectors of a are delocalized in the sense that begin aligned min limits i subset n i m vert mathbf v i vert ge frac m n log cn vert mathbf v vert end aligned for all m in log c n n comparing with related results in the range m in log c n n log c n in the i i d setting and with weaker probability estimates our lower bounds on vert mathbf v i vert strengthen an earlier estimate min nolimits i m vert mathbf v i vert ge c m n 6 vert mathbf v vert obtained in rudelson and vershynin geom funct anal 26 6 1716 1776 2016 and bounds min nolimits i m vert mathbf v i vert ge c m n 2 vert mathbf v vert in the real setting and min nolimits i m vert mathbf v i vert ge c m n 3 2 vert mathbf v vert in the complex setting established in luh and o rourke eigenvector delocalization for non hermitian random matrices and applications arxiv 1810 00489 as the case of real and complex gaussian matrices shows our bounds are optimal up to the polylogarithmic multiples we derive stronger estimates without the polylogarithmic error multiples for null vectors of real n 1 times n random matrices

local laws for polynomials of wigner matrices abstract we consider general self adjoint polynomials in several independent random matrices whose entries are centered and have the same variance we show that under certain conditions the local law holds up to the optimal scale i e the eigenvalue density on scales just above the eigenvalue spacing follows the global density of states which is determined by free probability theory we prove that these conditions hold for general homogeneous polynomials of degree two and for symmetrized products of independent matrices with i i d entries thus establishing the optimal bulk local law for these classes of ensembles in particular we generalize a similar result of anderson for anticommutator for more general polynomials our conditions are effectively checkable numerically

the spectral edge of some random band matrices we study the asymptotic distribution of the eigenvalues of random hermitian periodic band matrices focusing on the spectral edges the eigenvalues close to the edges converge in distribution to the airy point process if and only if the band is sufficiently wide w n 5 6 otherwise a different limiting distribution appears

isotropic self consistent equations for mean field random matrices we present a simple and versatile method for deriving an isotropic local laws for general random matrices constructed from independent random variables our method is applicable to mean field random matrices where all independent variables have comparable variances it is entirely insensitive to the expectation of the matrix in this paper we focus on the probabilistic part of the proof the derivation of the self consistent equations as a concrete application we settle in complete generality the local law for wigner matrices with arbitrary expectation

statistical properties of eigenvectors and eigenvalues of structured random matrices we study the eigenvalues and the eigenvectors of n x n structured random matrices of the form h w hw d with diagonal matrices d and w and h from the gaussian unitary ensemble using the supersymmetry technique we derive general asymptotic expressions for the density of states and the moments of the eigenvectors we find that the eigenvectors remain ergodic under very general assumptions but a degree of their ergodicity depends strongly on a particular choice of w and d for a special case of d 0 and random w we show that the eigenvectors can become critical and are characterized by non trivial fractal dimensions

interacting brownian particles and the wigner law in this paper we study interacting diffusing particles governed by the stochastic differential equationsdx j t n db j t d jon x 1 x n dt j 1 2 n here theb jare independent brownian motions in d and o n x 1 x n n i j v x i x j n i u x 1 the potentialv has a singularity at 0 strong enough to keep the particles apart and the potentialu serves to keep the particles from escaping to infinity our interest is in the behaviour as the number of particles increases without limit which we study through the empirical measure process we prove tightness of these processes in the case ofd 1 v x log x u x x 2 2 where it is possible to prove uniqueness of the limiting evolution and deduce that a limiting measure valued process exists this process is deterministic and converges to the wigner law ast some information on the rates of convergence is derived and the case of a cauchy initial distribution is analysed completely

uniform distribution of eigenfunctions on compact hyperbolic surfaces on demontre que les fonctions propres k du laplacien sur une surface hyperbolique compacte x devient uniformement distribuee sur x quand k

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

local spectral statistics of the addition of random matrices we consider the local statistics of h v x v u y u where v and u are independent haar distributed unitary matrices and x and y are deterministic real diagonal matrices in the bulk we prove that the gap statistics and correlation functions coincide with the gue in the limit when the matrix size n rightarrow infty under mild assumptions on x and certain rigidity assumptions on y the latter being an assumption on the convergence of the eigenvalues of y to the quantiles of its limiting spectral measure which we assume to have a density our method relies on running a carefully chosen diffusion on the unitary group and comparing the resulting eigenvalue process to dyson brownian motion our method also applies to the case when v and u are drawn from the orthogonal group our proof relies on the local law for h proved in bao et al commun math phys 349 3 947 990 2017 j funct anal 271 3 672 719 2016 adv math 319 251 291 2017 as well as the dbm convergence results of landon and yau commun math phys 355 3 949 1000 2017 and landon et al adv math 346 1137 1332 2019

a hereditarily indecomposable mathcal l _ infty space that solves the scalar plus compact problem we construct a hereditarily indecomposable banach space with dual space isomorphic to l1 every bounded linear operator on this space is expressible as i k with a scalar and k compact

stability of the matrix dyson equation and random matrices with correlations we consider real symmetric or complex hermitian random matrices with correlated entries we prove local laws for the resolvent and universality of the local eigenvalue statistics in the bulk of the spectrum the correlations have fast decay but are otherwise of general form the key novelty is the detailed stability analysis of the corresponding matrix valued dyson equation whose solution is the deterministic limit of the resolvent

fluctuations at the edges of the spectrum of the full rank deformed gue we consider a full rank deformation of the gue frac w n sqrt n a n where a n is a full rank hermitian matrix of size n and w n is a gue the empirical eigenvalue distribution mu a n of a n converges to a probability distribution nu we identify all the possible limiting eigenvalue statistics at the edges of the spectrum including outliers edges and merging points of connected components of the limiting spectrum the results are stated in terms of a deterministic equivalent of the empirical eigenvalue distribution of w n a n namely the free convolution of the semi circle distribution and the empirical eigenvalues distribution of a n

a generalized dynamic is lm model with delayed time in investment processes abstract in this paper we analyze a generalized dynamic is lm model with a government expenditure constraint and the capital accumulation equation that capital stock changes are caused by past investment decisions according to kalecki s idea of time delay in investment processes applying stability switch criteria we prove that the time delay causes the equilibrium point may lose or gain local stability so that the hopf bifurcation occurs

on the principal components of sample covariance matrices we introduce a class of m times m sample covariance matrices mathcal q which subsumes and generalizes several previous models the associated population covariance matrix sigma mathbb e mathcal q is assumed to differ from the identity by a matrix of bounded rank all quantities except the rank of sigma i m may depend on m in an arbitrary fashion we investigate the principal components i e the top eigenvalues and eigenvectors of mathcal q we derive precise large deviation estimates on the generalized components langle mathbf w varvec xi i rangle of the outlier and non outlier eigenvectors varvec xi i our results also hold near the so called bbp transition where outliers are created or annihilated and for degenerate or near degenerate outliers we believe the obtained rates of convergence to be optimal in addition we derive the asymptotic distribution of the generalized components of the non outlier eigenvectors a novel observation arising from our results is that unlike the eigenvalues the eigenvectors of the principal components contain information about the subcritical spikes of sigma the proofs use several results on the eigenvalues and eigenvectors of the uncorrelated matrix mathcal q satisfying mathbb e mathcal q i m as input the isotropic local marchenko pastur law established in bloemendal et al electron j probab 19 1 53 2014 level repulsion and quantum unique ergodicity of the eigenvectors the latter is a special case of a new universality result for the joint eigenvalue eigenvector distribution

anisotropic local laws for random matrices we develop a new method for deriving local laws for a large class of random matrices it is applicable to many matrix models built from sums and products of deterministic or independent random matrices in particular it may be used to obtain local laws for matrix ensembles that are anisotropic in the sense that their resolvents are well approximated by deterministic matrices that are not multiples of the identity for definiteness we present the method for sample covariance matrices of the form open image in new window where t is deterministic and x is random with independent entries we prove that with high probability the resolvent of q is close to a deterministic matrix with an optimal error bound and down to optimal spectral scales as an application we prove the edge universality of q by establishing the tracy widom airy statistics of the eigenvalues of q near the soft edges this result applies in the single cut and multi cut cases further applications include the distribution of the eigenvectors and an analysis of the outliers and bbp type phase transitions in finite rank deformations they will appear elsewhere we also apply our method to wigner matrices whose entries have arbitrary expectation i e we consider w a where w is a wigner matrix and a a hermitian deterministic matrix we prove the anisotropic local law for w a and use it to establish edge universality

localization and spectral statistics in a banded random matrix ensemble we investigate the localization properties of the eigenvectors of a banded random matrix ensemble in which the diagonal matrix elements increase along the diagonal we relate the results to a transition in the spectral statistics which is observed as a parameter is vaned and discuss the relevance of this model to the quantum mechanics of chaotic hamiltonian systems

the behaviour of eigenstates of arithmetic hyperbolic manifolds in this paper we study some problems arising from the theory of quantum chaos in the context of arithmetic hyperbolic manifolds we show that there is no strong localization scarring onto totally geodesic submanifolds arithmetic examples are given which show that the random wave model for eigenstates does not apply universally in 3 degrees of freedom

mass equidistribution for hecke eigenforms we prove a conjecture of rudnick and sarnak on the mass equidistribution of hecke eigenforms this builds upon independent work of the authors