branching rules for the supergroup su n m from those of su n m the decomposition of representations of supergroups into representations of subgroups is needed in practical applications in this paper we set up and exploit a fruitful one to one correspondence between the lie group branching su n m su n su m u 1 and the supergroup branchings su n m su n su m u 1 and su n1 n2 m1 m2 su n1 m1 su n2 m2 u 1 a simple and useful prescription is discovered for obtaining the su n m branching rules from those of su n m for any representation a large class of examples sufficient for many physical applications we can foresee are explicitly worked out and tabulated

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

clt for linear spectral statistics of large dimensional sample covariance matrices let bn 1 n t1 2nxnx nt1 2n where xn xij is n n with i i d complex standardized entries having finite fourth moment and t1 2n is a hermitian square root of the nonnegative definite hermitian matrix tn the limiting behavior as n with n n approaching a positive constant of functionals of the eigenvalues of bn where each is given equal weight is studied due to the limiting behavior of the empirical spectral distribution of bn it is known that these linear spectral statistics converges a s to a nonrandom quantity this paper shows their rate of convergence to be 1 n by proving after proper scaling that they form a tight sequence moreover if exppx211 0 and expp x11 4 2 or if x11 and tn are real and exppx411 3 they are shown to have gaussian limits

a clt for regularized sample covariance matrices we consider the spectral properties of a class of regularized estimators of large empirical covariance matrices corresponding to stationary but not necessarily gaussian sequences obtained by banding we prove a law of large numbers similar to that proved in the gaussian case by bickel and levina which implies that the spectrum of a banded empirical covariance matrix is an efficient estimator our main result is a central limit theorem in the same regime which to our knowledge is new even in the gaussian setup

wigner quantum systems two particles interacting via a harmonic potential i two dimensional space a noncanonical quantum system consisting of two nonrelativistic particles interacting via a harmonic potential is considered the center of mass position and momentum operators obey the canonical commutation relations whereas the internal variables are assumed to be the odd generators of the lie superalgebra sl 1 2 this assumption implies a set of constraints in the phase space which are explicitly written in the paper all finite dimensional irreducible representations of sl 1 2 are considered particular attention is paid to the physical representations i e the representations corresponding to hermitian position and momentum operators the properties of the physical observables are investigated in particular the operators of the internal hamiltonian the relative distance the internal momentum and the orbital momentum commute with each other the spectrum of these operators is finite the distance between the constituents is preserved in time it can take no more than three different values for any non negative integer or half integer l there exists a representation where the orbital momentum is l in units of 2 the position of any one of the particles cannot be localized since the operators of the coordinates do not commute with each other the constituents are smeared with a certain probability within a finite surface which moves with a constant velocity together with the center of mass

regularity conditions in the clt for linear eigenvalue statistics of wigner matrices abstract we show that the variance of centred linear statistics of eigenvalues of gue matrices remains bounded for large n for some classes of test functions less regular than lipschitz functions this observation is suggested by the limiting form of the variance which has previously been computed explicitly but it does not seem to appear in the literature we combine this fact with comparison techniques following tao vu and erdos yau et al and a littlewood paley type decomposition to extend the central limit theorem for linear eigenvalue statistics to functions in the holder class c 1 2 in the case of matrices of gaussian convolution type we also give a variance bound which implies the clt for test functions in the sobolev space h 1 and c 1 for general wigner matrices satisfying moment conditions if the additional assumption of the test function being supported away from the edge of the spectrum is made we prove the clt for test functions of regularity h 1 2 l and h 1 2 for gue and johansson matrices respectively previous results on the clt impose the existence and continuity of at least one classical derivative

finite dimensional irreducible representations of the lie superalgebra sl 1 3 in a gel fand zetlin basis a concept of a gel fand zetlin pattern for the lie superalgebra sl 1 3 is introduced within every finite dimensional irreducible sl 1 3 module the set of the gel fand zetlin patterns constitute an orthonormed basis called a gel fand zetlin basis expressions for the transformation of this basis under the action of the generators are written down for every finite dimensional irreducible representation

spectra of modular random graphs we compute spectra of symmetric random matrices defined on graphs exhibiting a modular structure modules are initially introduced as fully connected sub units of a graph by contrast inter module connectivity is taken to be incomplete two different types of inter module connectivities are considered one where the number of intermodule connections per node diverges and one where this number remains finite in the infinite module size limit in the first case the results can be understood as a perturbation of a superposition of semicircular spectral densities one would obtain for uncoupled modules in the second case matters can be more involved and depend in detail on inter module connectivities for suitable parameters we even find near triangular shaped spectral densities similar to those observed in certain scale free networks in a system consisting of just two coupled modules analytic results are presented for the infinite module size limit they are well corroborated by numerical simulations

the pearcey process the extended airy kernel describes the space time correlation functions for the airy process which is the limiting process for a polynuclear growth model the airy functions themselves are given by integrals in which the exponents have a cubic singularity arising from the coalescence of two saddle points in an asymptotic analysis pearcey functions are given by integrals in which the exponents have a quartic singularity arising from the coalescence of three saddle points a corresponding pearcey kernel appears in a random matrix model and a brownian motion model for a fixed time this paper derives an extended pearcey kernel by scaling the brownian motion model at several times and a system of partial differential equations whose solution determines associated distribution functions we expect there to be a limiting nonstationary process consisting of infinitely many paths which we call the pearcey process whose space time correlation functions are expressible in terms of this extended kernel

local spectral statistics of gaussian matrices with correlated entries we prove optimal local law bulk universality and non trivial decay for the off diagonal elements of the resolvent for a class of translation invariant gaussian random matrix ensembles with correlated entries

mixed supertableaux of the superunitary groups ii su n m the young supertableaux of the superunitary groups are studied we give the conditions on the size of legal supertableaux the type of atypicity for tensor representations which are irreducible making the link between the kac dynkin parameters and those of the supertableau a discussion is given on nonfully reducible representations and generalized atypical supertableaux

wigner quantum oscillators we present three groups of non canonical quantum oscillators the position and momentum operators of each group generate basic lie superalgebras namely sl 1 3 osp 1 6 and osp 3 2 the sl 1 3 oscillators have finite energy spectrum and finite dimensions the osp 1 6 oscillators are related to the para bose statistics the internal angular momentum s of the osp 3 2 oscillators takes no more than three half integer values in a particular representation s 1 2

operator norm consistent estimation of large dimensional sparse covariance matrices estimating covariance matrices is a problem of fundamental importance in multivariate statistics in practice it is increasingly frequent to work with data matrices x of dimension n x p where p and n are both large results from random matrix theory show very clearly that in this setting standard estimators like the sample covariance matrix perform in general very poorly in this large n large p setting it is sometimes the case that practitioners are willing to assume that many elements of the population covariance matrix are equal to 0 and hence this matrix is sparse we develop an estimator to handle this situation the estimator is shown to be consistent in operator norm when for instance we have p n as n oo in other words the largest singular value of the difference between the estimator and the population covariance matrix goes to zero this implies consistency of all the eigenvalues and consistency of eigenspaces associated to isolated eigenvalues we also propose a notion of sparsity for matrices that is compatible with spectral analysis and is independent of the ordering of the variables

representations of supergroups an explicit construction of representations of supergroups is given in terms of direct products of covariant and contravariant fundamental representations the rules of supersymmetrization are characterized by extended young supertableaux this constructive approach leads to explicit transformation properties of higher representations as well as to closed explicit formulas for characters from which other invariants such as dimensions and eigenvalues of all casimir operators can be calculated we have applied this approach so far to the supergroups su n m osp n 2m p n for which we have obtained all the representations constructible as direct products of the fundamental defining representations an argument is presented toward the irreducibility of all these representations

a law of large numbers for finite range dependent random matrices we consider random hermitian matrices in which distant above diagonal entries are independent but nearby entries may be correlated we find the limit of the empirical distribution of eigenvalues by combinatorial methods we also prove that the limit has an algebraic stieltjes transform by an argument based on dimension theory of noetherian local rings 2008 wiley periodicals inc

finite dimensional representations of the lie superalgebra sl 1 3 in a gel fand zetlin basis ii nontypical representations in a series of two papers all finite dimensional irreducible representations of the special linear lie superalgebra sl 1 3 are constructed explicit formulas for the generators in an orthonormal gel fand zetlin basis of the even subalgebra gl 3 are given this paper develops a background for constructing the representations expressions for the transformation properties of the basis under the action of the generators are written down within all typical sl 1 3 modules

second order freeness and fluctuations of random matrices i gaussian and wishart matrices and cyclic fock spaces abstract we extend the relation between random matrices and free probability theory from the level of expectations to the level of fluctuations we introduce the concept of second order freeness and interpret the global fluctuations of gaussian and wishart random matrices by a general limit theorem for second order freeness by introducing cyclic fock space we also give an operator algebraic model for the fluctuations of our random matrices in terms of the usual creation annihilation and preservation operators we show that orthogonal families of gaussian and wishart random matrices are asymptotically free of second order

central limit theorem for the heat kernel measure on the unitary group we prove that for a finite collection of real valued functions f1 fn on the group of complex numbers of modulus 1 which are derivable with lipschitz continuous derivative the distribution of trf1 trfn under the properly scaled heat kernel measure at a given time on the unitary group u n has gaussian fluctuations as n tends to infinity with a covariance for which we give a formula and which is of order n 1 in the limit where the time tends to infinity we prove that this covariance converges to that obtained by p diaconis and s n evans in a previous work on uniformly distributed unitary matrices finally we discuss some combinatorial aspects of our results

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

universality for general wigner type matrices we consider the local eigenvalue distribution of large self adjoint n times n random matrices mathbf h mathbf h with centered independent entries in contrast to previous works the matrix of variances s i j mathbbm e h i j 2 is not assumed to be stochastic hence the density of states is not the wigner semicircle law its possible shapes are described in the companion paper ajanki et al in quadratic vector equations on the complex upper half plane arxiv 1506 05095 we show that as n grows the resolvent mathbf g z mathbf h z 1 converges to a diagonal matrix mathrm diag mathbf m z where mathbf m z m 1 z dots m n z solves the vector equation 1 m i z z sum j s i j m j z that has been analyzed in ajanki et al quadratic vector equations on the complex upper half plane arxiv 1506 05095 we prove a local law down to the smallest spectral resolution scale and bulk universality for both real symmetric and complex hermitian symmetry classes

a clt for a band matrix model a law of large numbers and a central limit theorem are derived for linear statistics of random symmetric matrices whose on or above diagonal entries are independent but neither necessarily identically distributed nor necessarily all of the same variance the derivation is based on systematic combinatorial enumeration study of generating functions and concentration inequalities of the poincare type special cases treated with an explicit evaluation of limiting variances are generalized wigner and wishart matrices

spectra of modular and small world matrices we compute spectra of symmetric random matrices describing graphs with general modular structure and arbitrary inter and intra module degree distributions subject only to the constraint of finite mean connectivities we also evaluate spectra of a certain class of small world matrices generated from random graphs by introducing shortcuts via additional random connectivity components both adjacency matrices and the associated graph laplacians are investigated for the laplacians we find lifshitz type singular behaviour of the spectral density in a localized region of small values in the case of modular networks we can identify contributions of local densities of state from individual modules for small world networks we find that the introduction of short cuts can lead to the creation of satellite bands outside the central band of extended states exhibiting only localized states in the band gaps results for the ensemble in the thermodynamic limit are in excellent agreement with those obtained via a cavity approach for large finite single instances and with direct diagonalization results

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

real second order freeness and haar orthogonal matrices we demonstrate the asymptotic real second order freeness of haar distributed orthogonal matrices and an independent ensemble of random matrices our main result states that if we have two independent ensembles of random matrices with a real second order limit distribution and one of them is invariant under conjugation by an orthogonal matrix then the two ensembles are asymptotically real second order free this captures the known examples of asymptotic real second order freeness introduced by redelmeier

poincare maps of duffing type oscillators and their reduction to circle maps ii methods and numerical results bifurcation diagrams and plots of lyapunov exponents in the r plane for duffing type oscillators exhibit a regular pattern of repeating self similar tongues with complex internal structure we demonstrate here how this behaviour is easily understood qualitatively and quantitatively from a poincare map of the system in action angle variables in the limit of large driving force or equivalently small driving frequency this map approaches the one dimensional form as derived in paper i this second paper describes our approach to calculating the various constants and functions introduced in paper i it gives numerical applications of the theory and tests its range of validity by comparison with results from the numerical integration of duffing type equations finally we show how to extend the range in the parameter space where the map is applicable

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

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

canonical realizations of lie superalgebras ladder representations of the lie superalgebra a m n a simple formula for realizations of lie superalgebras in terms of bose and fermi creation and annihilation operators is given the essential new feature is that bose and fermi operators mutually anticommute the fock representation of these operators is used in order to construct a class of irreducible finite dimensional representations of the simple lie superalgebra a m n the matrix elements of the generators are written down for m 0 all representations turn out to be nontypical

an answer to littlewood s problem on boundedness for super linear duffing s equations we consider the duffing s equation x g x p t where p t is a continuous and periodic function of t in r and g x is a continuously differentiable function of x in r with the super linear condition g x x as x the purpose of this paper is to prove the following theorem which is an answer to littlewood s boundedness problem for super linear duffing s equations theorem every solution of is bounded for t inr

absolutely continuous spectrum for the anderson model on a tree a geometric proof of klein s theorem we give a new proof of a version of klein s theorem on the existence of absolutely continuous spectrum for the anderson model on the bethe lattice at weak disorder

oscillatory behavior of solutions of forced second order differential equations with alternating coefficients abstract in this paper we study the oscillatory behavior of solutions of the forced second order nonlinear ordinary functional differential equation x t g x t p t d dt sufficient conditions are established for all solutions of the equation to be oscillatory as well as bounded

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

diagram and superfield techniques in the classical superalgebras introduces the concept of graded permutation group in the analysis of tensor operators in the classical superalgebras for u m n and su m n irreducible tensor representations correspond to classes of young tableaux with definite graded symmetry type diagram techniques are given for kronecker products dimensions and branching rules the tensor techniques are complemented by the introduction of a superfluid formalism in which u m n and su m n act on polynomial functions over the appropriate superspace such superfluids may admit constraints a general superfield interpolates between the classes of young tableaux which correspond to particular types of constraint the tensor and superfield techniques are illustrated with case studies of su 2 1 and su m 1

integral averaging techniques for the oscillation of second order nonlinear differential equations abstract in this paper we enlist some known and establish some new oscillation criteria for the second order nonlinear differential equations of the form if a t x t t su p t t q t f x t 0 where the coefficients p t and q t are not assumed to be nonnegative for all large values of t these criteria are obtained by using integral averaging techniques and can be applied to some special cases where other classical oscillation results are not applicable a systematic study is attempted to extend improve and correlate a number of existing results

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

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

second order cumulants of products we derive a formula which expresses a second order cumulant whose entries are products as a sum of cumulants where the entries are single factors this extends to the second order case the formula of krawczyk and speicher we apply our result to the problem of calculating the second order cumulants of a semi circular and haar unitary operator

asymptotic properties of large random matrices with independent entries we study the normalized trace gn z n 1 tr h zi 1 of the resolvent of n n real symmetric matrices h 1 jk wjk n j k 1n assuming that their entries are independent but not necessarily identically distributed random variables we develop a rigorous method of asymptotic analysis of moments of gn z for iz 0 where 0 is determined by the second moment of wjk by using this method we find the asymptotic form of the expectation e gn z and of the connected correlator e gn z1 gn z2 e gn z1 e gn z2 we also prove that the centralized trace ngn z e ngn z has the gaussian distribution in the limit n based on these results we present heuristic arguments supporting the universality property of the local eigenvalue statistics for this class of random matrix ensembles

universal singularity at the closure of a gap in a random matrix theory we consider a hamiltonian h h 0 v in which h 0 is a given non random hermitian matrix and v is an n times n hermitian random matrix with a gaussian probability distribution we had shown before that dyson s universality of the short range correlations between energy levels holds at generic points of the spectrum independently of h 0 we consider here the case in which the spectrum of h 0 is such that there is a gap in the average density of eigenvalues of h which is thus split into two pieces when the spectrum of h 0 is tuned so that the gap closes a new class of universality appears for the energy correlations in the vicinity of this singular point