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

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 of level correlation function of sparse random matrices the statistical properties of sparse random matrices ensembles are investigated by means of a supersymmetric approach with the use of a functional generalization of the hubbard stratonovich transformation when used to calculate the density of states the method is shown to be absolutely equivalent to the replica trick the model turns out to bear a close resemblance to the anderson model on the bethe lattice it possesses a delocalization transition that occurs with an increase in the mean connectivity parameter in the delocalized phase the level level correlation function proves to have a universal dyson form with the full density of states replaced by the contribution from the infinite cluster

bulk universality of general β ensembles with non convex potential we prove the bulk universality of the ensembles with non convex regular analytic potentials for any 0 this removes the convexity assumption appeared in the earlier work p bourgade l erdos and h t yau universality of general ensembles preprint arxiv 0907 5605 2011 the convexity condition enabled us to use the logarithmic sobolev inequality to estimate events with small probability the new idea is to introduce a convexified measure so that the local statistics are preserved under this convexification

quantized output feedback control for networked control systems this paper addresses the problem of output feedback control for networked control systems ncss with limited communication capacity firstly we propose a new model to describe the non ideal network conditions and the input output state quantization of the ncss in a unified framework secondly based on our newly proposed model and an improved separation lemma the observer based controller is developed for the asymptotical stabilization of the ncss which are shown in terms of nonlinear matrices inequalities the nonlinear problems can be computed through solving a convex optimization problems and the observed and controller gains could be derived by solving a set of linear matrix inequalities thirdly two simulation examples are given to demonstrate the effectiveness of the proposed method

decentralized stabilization of large scale systems with multiple delays in the interconnections in this paper the problem of robust stabilization of a class of time varying large scale systems subject to external disturbances uncertain parameters and multiple time varying delays in the interconnections and non linear inputs is considered linear memory less controllers are developed to guarantee that all solutions of the class of systems considered are uniformly ultimately bounded also non linear memoryless controllers are synthesized to guarantee that the solutions are uniformly ultimately bounded with respect to a bound with any prespecified convergence rate towards it finally an example is included to illustrate the results developed in this paper

on asymptotic behavior of multilinear eigenvalue statistics of random matrices we prove the law of large numbers and the central limit theorem for analogs of u and v von mises statistics of eigenvalues of random matrices as their size tends to infinity we show first that for a certain class of test functions kernels determining the statistics the validity of these limiting laws reduces to the validity of analogous facts for certain linear eigenvalue statistics we then check the conditions of the reduction statements for several most known ensembles of random matrices the reduction phenomenon is well known in statistics dealing with i i d random variables it is of interest that an analogous phenomenon is also the case for random matrices whose eigenvalues are strongly dependent even if the entries of matrices are independent

transport maps for beta matrix models and universality we construct approximate transport maps for non critical beta matrix models that is maps so that the push forward of a non critical beta matrix model with a given potential is a non critical beta matrix model with another potential up to a small error in the total variation distance one of the main features of our construction is that these maps enjoy regularity estimates that are uniform in the dimension in addition we find a very useful asymptotic expansion for such maps which allows us to deduce that local statistics have the same asymptotic behavior for both models

robust exponential stability of uncertain systems with time varying delays focuses on the problem of robust exponential stability of a class of uncertain systems described by functional differential equations with time varying delays the uncertainties are assumed to be continuous time varying nonlinear and norm bounded sufficient conditions for robust exponential stability are given for both single and multiple delays cases

adaptive regulation for a class of uncertain systems with partial knowledge of uncertainty bounds an adaptive regulation method is presented for a class of uncertain systems where the uncertainty bounds are partially known both the state and the parameter error of the closed loop system are shown to be ultimately bounded the proposed method improves on the adaptive control method of m j corless and g leitmann 1981 so that performance degradation and the destabilization of the control system can be effectively presented in practical situations

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

design of a unified adaptive fuzzy observer for uncertain nonlinear systems this paper presents an adaptive fuzzy observer for a class of uncertain nonlinear systems more precisely we propose a unified approach for designing such an observer with some design flexibility so that it can be easily adaptable and employed either as a high gain or a sliding mode observer by selecting its gain appropriately additionally we derive a suitable parameter adaptation law so that the proposed observer is robust with respect to ubiquitous fuzzy approximation errors and external disturbances we also show that the observation error is ultimately bounded using a lyapunov approach without having recourse to the usual strictly positive real spr condition or a suitable observation error filtering the effectiveness of the proposed observers is illustrated through two simulation case studies taken from the adaptive fuzzy control literature

the nevai condition and a local law of large numbers for orthogonal polynomial ensembles we consider asymptotics of orthogonal polynomial ensembles in the macroscopic and mesoscopic scales we prove both global and local laws of large numbers analogous to the recently proven local semicircle law for wigner matrices under fairly weak conditions on the underlying measure our main tools are a general concentration inequality for determinantal point processes with a kernel that is a selfadjoint projection and a strengthening of the nevai condition from the theory of orthogonal polynomials

technical communique stabilization of a class of uncertain large scale stochastic systems with time delays this paper focuses on a class of large scale interconnected bilinear stochastic systems with time delays and time varying parameter uncertainties the problem we address is to design a robust decentralized controller such that the closed loop interconnected stochastic system is globally asymptotically stable in probability for all admissible uncertainties and time delays first robust stability analysis is given in terms of a set of linear matrix inequalities lmis then it is shown that the robust decentralized stabilization can be solved by an lmi based design method

robust stability and stabilizing controller design of fuzzy systems with discrete and distributed delays in this paper we consider delay dependent stability conditions of takagi sugeno fuzzy systems with discrete and distributed delays although many kinds of stability conditions for fuzzy systems with discrete delays have already been obtained almost no stability condition for fuzzy systems with distributed delays has appeared in the literature this is also true in case of the robust stability for uncertain fuzzy systems with distributed delays here we employ a generalized lyapunov functional to obtain delay dependent stability conditions of fuzzy systems with discrete and distributed delays we introduce some free weighting matrices to such a lyapunov functional in order to reduce the conservatism in stability conditions these techniques lead to generalized and less conservative stability conditions we also consider the robust stability of fuzzy time delay systems with uncertain parameters applying the same techniques made on the stability conditions we obtain delay dependent sufficient conditions for the robust stability of uncertain fuzzy systems with discrete and distributed delays moreover we consider the state feedback stabilization based on stability and robust stability conditions we obtain conditions for the state feedback controller to stabilize the fuzzy time delay systems finally we give two examples to illustrate our results delay dependent stability conditions obtained here are shown to guarantee a wide stability region

the normal bundle of a curve on a quadric it was some years ago that grauert posed the problem to find an example of a smooth space curve whose normal bundle is indecomposable the first to produce such an example was van de ven 3 he studied space curves c of degree 5 and genus 2 such a curve lies necessarily on an irreducible quadric q if q is smooth then c has bidegree 2 3 or 3 2 and its normal bundle n c does not split if however q is a quadric cone then n c does in fact split the whole purpose of this note is to point out that the situation described by van de ven is typical for curves on quadrics we shall prove

decentralized adaptive robust control for a class of large scale systems including delayed state perturbations in the interconnections the problem of decentralized stabilization is considered for a class of large scale time varying systems with delayed state perturbations in the interconnections in this note the upper bounds of the uncertainties in the interconnections are assumed to be unknown the adaptation laws are proposed to estimate such unknown bounds and by making use of their updated values a class of decentralized memoryless state feedback controllers is constructed based on lyapunov stability theory and lyapunov krasovskii functional it is shown that the solutions to the resulting adaptive closed loop large scale time delay system can be guaranteed to be uniformly ultimately bounded finally a numerical example is given to demonstrate the validity of the results

on the normal bundles of smooth rational space curves in this note we consider smooth rational curves c of degree n in threedimensional projective space ip 3 over a closed field of characteristic 0 to avoid trivial exceptions we shall always assume that n 4 this does not hold however for certain auxiliary curves we shall consider let n n c be the normal bundle of c in ip 3 since degel ip3 4 and d e g c l l p 0 2 we have that d e g c l n 4 n 2 by a well known theorem of grothendieck the bundle n is a direct sum of two line bundles hence n o c 2 n l a g o c 2 n 1 a for some non negative a a c which is uniquely determined by c the question we would like to answer is an obvious one which values of a occur we shall show theorem 4 below that a value a occurs if and only if 0 0 therefore hi c n o it follows k p 150 that c represents a smooth point on the chow variety ch 3 1 n of effective cycles of dimension 1 and degree n in ip 3 since the set of all smooth rational curves with a fixed degree is obviously connected we see that the smooth c s represent a smooth irreducible 4n dimensional zariski open subset s of ch 3 1 n in a for thcoming paper e v we shall prove the following

orthogonal and symplectic matrix models universality and other properties we study orthogonal and symplectic matrix models with polynomial potentials and multi interval supports of the equilibrium measures for these models we find the bounds similar to those for the hermitian matrix models for the rate of convergence of linear eigenvalue statistics and for the variance of linear eigenvalue statistics and find the logarithms of partition functions up to the order o 1 we prove also the universality of local eigenvalue statistics in the bulk

normal bundle of rational curves and waring decomposition abstract the problem of determining the splitting of the normal bundle of rational space curves has been considered in the 80s in a series of papers by ghione and sacchiero and by eisenbud and van de ven with our approach we are able to obtain results for curves embedded in p m for m 3 and we find an interesting interplay between the waring decomposition of binary forms and the splitting of the normal bundle

the singular points of a rational plane curve and its restricted tangent bundle abstract we consider the vector bundle t p 2 where denotes a generically one to one parametrization of a rational curve d of degree d d 2 in p 2 we study a parameter a d that controls the splitting of t p 2 as a direct sum of line bundles we find that specific properties of d related to its singular points characterize a d these properties are multiplicity non collinearity of triples and construction of a curve of degree strictly smaller than d d with multiplicity bounded by the multiplicity of d at corresponding points

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

observer based adaptive control for uncertain time delay systems in this paper we will focus on investigating the observer based controlling problem of time delay systems first we investigate a class of simple time delay systems the corresponding adaptive observer and controller are designed which are both independent of the time delays based on lyapunov stability theory we prove that the closed loop system is asymptotically stable next we further consider the interconnected time delay system case the corresponding adaptive observer and controller are designed we prove that the resulting closed loop system is also asymptotically stable simulations on controlling time delay systems and interconnected systems are investigated and the results show that the designed controllers are feasible and efficient

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

a note on the eigenvalue density of random matrices the distribution of eigenvalues of n n random matrices in the limit n is the solution to a variational principle that determines the ground state energy of a confined fluid of classical unit charges this fact is a consequence of a more general theorem proven here in the statistical mechanics of unstable interactions our result establishes the eigenvalue density of some ensembles of random matrices which were not covered by previous theorems

on universality of local edge regime for the deformed gaussian unitary ensemble we consider the deformed gaussian ensemble h n h n 0 m n in which h n 0 is a hermitian matrix possibly random and m n is the gaussian unitary random matrix gue independent of h n 0 assuming that the normalized counting measure of h n 0 converges weakly in probability if random to a non random measure n 0 with a bounded support and assuming some conditions on the convergence rate we prove the universality of the local eigenvalue statistics near the edge of the limiting spectrum of h n

asymptotic behavior of the verblunsky coefficients for the opuc with a varying weight we present an asymptotic analysis of the verblunsky coefficients for the polynomials orthogonal on the unit circle with the varying weight e nv cos x assuming that the potential v has four bounded derivatives on 1 1 and the equilibrium measure has a one interval support we obtain the asymptotics as a solution of the system of string equations

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

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

edge universality for orthogonal ensembles of random matrices we prove edge universality of local eigenvalue statistics for orthogonal invariant matrix models with real analytic potentials and one interval limiting spectrum our starting point is the result of shcherbina commun math phys 285 957 974 2009 on the representation of the reproducing matrix kernels of orthogonal ensembles in terms of scalar reproducing kernel of corresponding unitary ensemble

adaptive stabilizing state feedback controllers of uncertain dynamical systems with multiple time delays the problem of robust stabilization for a class of uncertain dynamical systems with multiple delayed state perturbations is considered in this paper it is assumed that each perturbation is bounded by a linear function of delayed state with unknown gains and an adaptation law is proposed to estimate these unknown gains moreover by making use of the updated values of these unknown bounds we propose a memoryless state feedback controller for such a class of uncertain time delay systems based on lyapunov stability theory and lyapunov krasovskii functional it is shown that the closed loop dynamical system resulting from the proposed adaptive robust control schemes is globally stable in the sense of uniform ultimate boundedness finally a numerical example is given to demonstrate the validity of the results

stabilization of uncertain dynamic systems including state delay the authors consider a class of time varying systems with time varying state delay and uncertain input and parameters they present sufficient conditions on the open loop to obtain an arbitrary degree closed loop stability they construct a min max controller from knowledge of the upper bound of the delay

log gases on quadratic lattices via discrete loop equations and q boxed plane partitions abstract we study a general class of log gas ensembles on shifted quadratic lattices we prove that the corresponding empirical measures satisfy a law of large numbers and that their global fluctuations are gaussian with a universal covariance we apply our general results to analyze the asymptotic behavior of a q boxed plane partition model introduced by borodin gorin and rains in particular we show that the global fluctuations of the height function on a fixed slice are described by a one dimensional section of a pullback of the two dimensional gaussian free field our approach is based on a q analogue of the schwinger dyson or loop equations which originate in the work of nekrasov and his collaborators and extends the methods developed by borodin gorin and guionnet to quadratic lattices

on universality for orthogonal ensembles of random matrices we prove universality of local eigenvalue statistics in the bulk of the spectrum for orthogonal invariant matrix models with real analytic potentials with one interval limiting spectrum our starting point is the tracy widom formula for the matrix reproducing kernel the key idea of the proof is to represent the differentiation operator matrix written in the basis of orthogonal polynomials as a product of a positive toeplitz matrix and a two diagonal skew symmetric toeplitz matrix

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

universality limits for generalized jacobi measures abstract in this paper universality limits are studied in connection with measures which exhibit power type singular behavior somewhere in their support we extend the results of lubinsky for jacobi measures supported on 1 1 to generalized jacobi measures supported on a compact subset of the real line where the singularity can be located in the interior or at an endpoint of the support the analysis is based upon the riemann hilbert method christoffel functions the polynomial inverse image method of totik and the normal family approach of lubinsky

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

fluctuations of matrix elements of regular functions of gaussian random matrices we find the limit of the variance and prove the central limit theorem clt for the matrix elements jk m j k 1 n of a regular function of the gaussian matrix m goe and gue as its size n tends to infinity we show that unlike the linear eigenvalue statistics tr m a traditional object of random matrix theory whose variance is bounded as n and the clt is valid for tr m e tr m the variance of jk m is o 1 n and the clt is valid for sqrt n varphi jk m mathbf e varphi jk m this shows the role of eigenvectors in the forming of the asymptotic regime of various functions statistics of random matrices our proof is based on the use of the fourier transform as a basic characteristic function unlike the stieltjes transform and moments used in majority of works of the field we also comment on the validity of analogous results for other random matrices

on universality of bulk local regime of the hermitian sample covariance matrices we consider the hermitian sample covariance matrices hn m 1 n1 2am nam n n1 2 in which n is a positive definite hermitian matrix possibly random and am n is a n m complex gaussian random matrix independent of n and m n such that mn 1 c 1 assuming that the normalized counting measure of n converges weakly in probability to a nonrandom measure n 0 with a bounded support we prove the universality of the local eigenvalue statistics in the bulk of the limiting spectrum of hn

betti numbers for fat point ideals in the plane a geometric approach we consider the open problem of determining the graded betti numbers for fat point subschemes z supported at general points of p 2 we relate this problem to the open geometric problem of determining the splitting type of the pullback of p2 to the normalization of certain rational plane curves we give a conjecture for the graded betti numbers which would determine them in all degrees but one for every fat point subscheme supported at general points of p 2 we also prove our betti number conjecture in a broad range of cases an appendix discusses many more cases in which our conjecture has been verified computationally and provides a new and more efficient computational approach for computing graded betti numbers in certain degrees it also demonstrates how to derive explicit conjectural values for the betti numbers and how to compute splitting types

the law of large numbers for the maximum of almost gaussian log correlated fields coming from random matrices we compute the leading asymptotics as n rightarrow infty of the maximum of the field q n q log q a n q in mathbb c for any unitarily invariant hermitian random matrix a n associated to a non critical real analytic potential hence we verify the leading order in a conjecture of fyodorov and simm nonlinearity 29 2837 2016 arxiv 1503 07110 math ph formulated for the gue the method relies on a classical upper bound and a more sophisticated lower bound based on a variant of the second moment method which exploits the hyperbolic branching structure of the field q n q q in mathbb h specifically we compare q n to an idealized gaussian field by means of exponential moments in principle this method could also be applied to random fields coming from other point processes provided that one can compute certain mixed exponential moments for unitarily invariant ensembles we show that these assumptions follow from the fyodorov strahov formula fyodorov and strahov in j phys a 36 12 3203 3213 2003 https doi org 10 1088 0305 4470 36 12 320 and asymptotics of orthogonal polynomials derived in deift et al commun pure appl math 52 11 1335 1425 1999 https doi org 10 1002 sici 1097 0312 199911 52 11 3 0 co 2 1

an observer based approach to controlling time delay chaotic systems via takagi sugeno fuzzy model this study presents a fuzzy model based control criterion for time delay chaotic systems with uncertainty the takagi sugeno t s fuzzy model is adopted for representing a chaotic system possessing a time delay a unified approach for designing a fuzzy observer which integrates parallel distributed compensation with an adaptive updating method is first discussed an observer based fuzzy control scheme is then developed to deal with stabilization and tracking problems based on lyapunov stability analysis sufficient conditions that ensure robust control performance of time delay chaotic systems are derived these conditions are represented in terms of linear matrix inequalities and adaptive updating laws finally the proposed approach is validated through numerical synchronization stabilization and model tracking control simulations

bulk universality and related properties of hermitian matrix models we give a new proof of universality properties in the bulk of spectrum of the hermitian matrix models assuming that the potential that determines the model is globally c 2 and locally c 3 function see theorem 3 1 the proof as our previous proof in pastur and shcherbina in j stat phys 86 109 147 1997 is based on the orthogonal polynomial techniques but does not use asymptotics of orthogonal polynomials rather we obtain the sin kernel as a unique solution of a certain non linear integro differential equation that follows from the determinant formulas for the correlation functions of the model we also give a simplified and strengthened version of paper boutet de monvel et al in j stat phys 79 585 611 1995 on the existence and properties of the limiting normalized counting measure of eigenvalues we use these results in the proof of universality and we believe that they are of independent interest

the single ring theorem we study the empirical measure lan of the eigenvalues of non normal square matrices of the form an untnvn with un vn independent haar distributed on the unitary group and tn real diagonal we show that when the empirical measure of the eigenvalues of tn converges and tn satisfies some technical conditions lan converges towards a rotationally invariant measure on the complex plane whose support is a single ring in particular we provide a complete proof of feinberg zee single ring theorem 6 we also consider the case where un vn are independent haar distributed on the orthogonal group