exponential bounds for the support convergence in the single ring theorem abstract we consider an n n matrix of the form a u t v with u v some independent haar distributed unitary matrices and t a deterministic matrix we prove that for k n 1 6 and b 2 1 n tr t 2 as n tends to infinity we have e tr a k a k b 2 k and e tr a k 2 b 2 k this gives a simple proof with slightly weakened hypothesis of the convergence of the support in the single ring theorem improves the available error bound for this convergence from n to e c n 1 6 and proves that the rate of this convergence is at most n 1 6 log n

computing orthogonal rational functions with poles near the boundary computing orthogonal rational functions is a far from trivial problem especially for poles close to the boundary of the support of the orthogonality measure in this paper we analyze some of the difficulties involved and present two different approaches for solving this problem

orthogonal rational functions for system identification numerical aspects recently there has been a growing interest in the use of orthogonal rational functions orfs in system identification there are many advantages over more classical techniques probably due to a known explicit expression for the basis functions when the orthogonality weight is uniformly equal to 1 the so called malmquist basis the attention has been on the development of methods using this basis however for some discrete identification problems this choice of the orthogonality weight may still lead to serious numerical problems due to the ill conditioning of the linear system of equations to be solved in this note we give an algorithm based on a more general system of orf to overcome the numerical problem and which allows for a fast order update of the estimate

orthogonal basis functions in discrete least squares rational approximation we consider a problem that arises in the field of frequency domain system identification if a discrete time system has an input output relation y z g z u z with transter function g then the problem is to find a rational approximation n for g the data given are measurements of input and output spectra in the frequency points zk u zk y zk nk 1 together with some weight the approximation criterion is to minimize the weighted discrete least squares norm of the vector obtained by evaluating g n in the measurement points if the poles of the system are fixed then the problem reduces to a linear least squares problem in two possible ways by multiplying out the denominators and hide these in the weight which leads to the construction of orthogonal vector polynomials or the problem can be solved directly using an orthogonal basis of rational functions the orthogonality of the basis is important because if the transfer function n is represented with respect to a nonorthogonal basis then this least squares problem can be very ill conditioned even if an orthogonal basis is used but with respect to the wrong inner product e g the lebesgue measure on the unit circle numerical instability can be fatal in practice we show that both approaches lead to an inverse eigenvalue problem which forms the common framework in which fast and numerically stable algorithms can be designed for the computation of the orthonormal basis

limit laws for random matrices and free products in earlier articles we studied a kind of probability theory in the framework of operator algebras with the tensor product replaced by the free product we prove here that free random variables naturally arise as limits of random matrices and that wigner s semicircle law is a consequence of the central limit theorem for free random variables in this way we obtain a non commutative limit distribution of a general gaussian random matrix as an operator in a certain operator algebra wigner s law being given by the trace of the spectral measure of the selfadjoint component of this operator

system identification with generalized orthonormal basis functions a least squares identification method is studied that estimates a finite number of expansion coefficients in the series expansion of a transfer function where the expansion is in terms of recently introduced generalized basis functions the basis functions are orthogonal in h2 and generalize the pulse laguerre and kautz bases one of their important properties is that when chosen properly they can substantially increase the speed of convergence of the series expansion this leads to accurate approximate models with only a few coefficients to be estimated explicit bounds are derived for the bias and variance errors that occur in parameter estimates as well as in the resulting transfer function estimates

a new application of random matrices ext c _ mathrm red f_2 is not a group in the process of developing the theory of free probability and free entropy voiculescu introduced in 1991 a random matrix model for a free semicircular system since then random matrices have played a key role in von neumann algebra theory cf v8 v9 the main result of this paper is the following extension of voiculescui s random matrix result let x n 1 x n r be a system of r stochastically independent n i n gaussian self adjoint random matrices as in voiculescui s random matrix paper v4 and let x1 xr be a semi circular system in a c probability space then for every polynomial p in r noncommuting variables lim niaeia p x n 1 o x n r o p x1 xr for almost all o in the underlying probability space we use the result to show that the ext invariant for the reduced c algebra of the free group on 2 generators is not a group but only a semi group this problem has been open since anderson in 1978 found the first example of a c algebra a for which ext a is not a group

system identification using kautz models in this paper the problem of approximating a linear time invariant stable system by a finite weighted sum of given exponentials is considered system identification schemes using laguerre models are extended and generalized to kautz models which correspond to representations using several different possible complex exponentials in particular linear regression methods to estimate this sort of model from measured data are analyzed the advantages of the proposed approach are the simplicity of the resulting identification scheme and the capability of modeling resonant systems using few parameters the subsequent analysis is based on the result that the corresponding linear regression normal equations have a block toeplitz structure several results on transfer function estimation are extended to discrete kautz models for example asymptotic frequency domain variance expressions

a generalized orthonormal basis for linear dynamical systems in many areas of signal system and control theory orthogonal functions play an important role in issues of analysis and design in this paper it is shown that there exist orthogonal functions that in a natural way are generated by stable linear dynamical systems and that compose an orthonormal basis for the signal space l sub 2sup n to this end use is made of balanced realizations of inner transfer functions the orthogonal functions can be considered as generalizations of for example the pulse functions laguerre functions and kautz functions and give rise to an alternative series expansion of rational transfer functions it is shown how we can exploit these generalized basis functions to increase the speed of convergence in a series expansion i e to obtain a good approximation by retaining only a finite number of expansion coefficients consequences for identification of expansion coefficients are analyzed and a bound is formulated on the error that is made when approximating a system by a finite number of expansion coefficients

outlier eigenvalues for deformed i i d random matrices we consider a square random matrix of size n of the form a y where a is deterministic and y has iid entries with variance 1 n under mild assumptions as n grows the empirical distribution of the eigenvalues of a y converges weakly to a limit probability measure beta on the complex plane this work is devoted to the study of the outlier eigenvalues i e eigenvalues in the complement of the support of beta even in the simplest cases a variety of interesting phenomena can occur as in earlier works we give a sufficient condition to guarantee that outliers are stable and provide examples where their fluctuations vary with the particular distribution of the entries of y or the jordan decomposition of a we also exhibit concrete examples where the outlier eigenvalues converge in distribution to the zeros of a gaussian analytic function

outliers in the single ring theorem this text is about spiked models of non hermitian random matrices more specifically we consider matrices of the type mathbf a mathbf p where the rank of mathbf p stays bounded as the dimension goes to infinity and where the matrix mathbf a is a non hermitian random matrix satisfying an isotropy hypothesis its distribution is invariant under the left and right actions of the unitary group the macroscopic eigenvalue distribution of such matrices is governed by the so called single ring theorem due to guionnet krishnapur and zeitouni we first prove that if mathbf p has some eigenvalues out of the maximal circle of the single ring then mathbf a mathbf p has some eigenvalues called outliers in the neighborhood of those of mathbf p which is not the case for the eigenvalues of mathbf p in the inner cycle of the single ring then we study the fluctuations of the outliers of mathbf a around the eigenvalues of mathbf p and prove that they are distributed as the eigenvalues of some finite dimensional random matrices such kind of fluctuations had already been shown for hermitian models more surprising facts are that outliers can here have very various rates of convergence to their limits depending on the jordan canonical form of mathbf p and that some correlations can appear between outliers at a macroscopic distance from each other a fact already noticed by knowles and yin in ann probab 42 1980 2031 2014 in the hermitian case but only for non gaussian models whereas spiked gaussian matrices belong to our model and can have such correlated outliers our first result generalizes a result by tao proved specifically for matrices with i i d entries whereas the second one about the fluctuations is new

a limit theorem at the edge of a non hermitian random matrix ensemble the study of the edge behaviour in the classical ensembles of gaussian hermitian matrices has led to the celebrated distributions of tracy widom here we take up a similar line of inquiry in the non hermitian setting we focus on the family of n n random matrices with all entries independent and distributed as complex gaussian of mean zero and variance 1 n this is a fundamental non hermitian ensemble for which the eigenvalue density is known using this density our main result is a limit law for the scaled spectral radius as n as a corollary we get the analogous statement for the case where the complex gaussians are replaced by quaternion gaussians

frequency domain identification with generalized orthonormal basis functions a method is considered for the identification of linear parametric models based on a least squares identification criterion that is formulated in the frequency domain to this end use is made of the empirical transfer function estimate etfe identified from time domain data as a parametric model structure use is made of a finite expansion sequence in terms of recently introduced generalized basis functions being generalizations of the classical pulse and laguerre and kautz types of bases an asymptotic analysis of the estimated models is provided and conditions for consistency are formulated explicit and transparent bias and variance expressions are established the latter ones also valid in a situation of undermodeling

system identification using laguerre models the traditional approach of expanding transfer functions and noise models in the delay operator to obtain linear in the parameters predictor models leads to approximations of very high order in cases of rapid sampling and or dispersion in time constants by using prior information about the time constants of the system more appropriate expansions related to laguerre networks are introduced and analyzed it is shown that the model order can be reduced compared to arx fir ar modeling by using laguerre models furthermore the numerical accuracy of the corresponding linear regression estimation problem is improved by a suitable choice of the laguerre parameter consistency error bounds persistence of excitation conditions and asymptotic statistical properties are investigated this analysis is based on the result that the covariance matrix of the regression vector of a laguerre model has a toeplitz structure

the first passage sets of the 2d gaussian free field convergence and isomorphisms in a previous article we introduced the first passage set fps of constant level a of the two dimensional continuum gaussian free field gff on finitely connected domains informally it is the set of points in the domain that can be connected to the boundary by a path along which the gff is greater than or equal to a this description can be taken as a definition of the fps for the metric graph gff and in the current article we prove that the metric graph fps converges towards the continuum fps in the hausdorff distance we also draw numerous consequences in particular we obtain a relatively simple proof of the fact that certain natural interfaces of the metric graph gff converge to hbox sle 4 level lines these results improve our understanding of the continuum gff by strengthening its relationship with the critical brownian loop soup indeed a new construction of the fps using clusters of brownian loops and excursions helps to strengthen the known gff isomorphism theorems and allows us to use brownian loop soup techniques to prove technical results on the geometry of the gff we also obtain a new representation of brownian loop soup clusters and as a consequence we prove that the clusters of a critical brownian loop soup admit a non trivial minkowski content in the gauge r mapsto log r 1 2 r 2

invertibility of random matrices unitary and orthogonal perturbations author s rudelson mark vershynin roman abstract we show that a perturbation of any fixed square matrix d by a random unitary matrix is well invertible with high probability a similar result holds for perturbations by random orthogonal matrices the only notable exception is when d is close to orthogonal as an application these results completely eliminate a hard to check condition from the single ring theorem by guionnet krishnapur and zeitouni

the fundamental role of general orthonormal bases in system identification the purpose of this paper is threefold firstly it is to establish that contrary to what might be expected the accuracy of well known and frequently used asymptotic variance results can depend on choices of fixed poles or zeros in the model structure secondly it is to derive new variance expressions that can provide greatly improved accuracy while also making explicit the influence of any fixed poles or zeros this is achieved by employing certain new results on generalized fourier series and the asymptotic properties of toeplitz like matrices in such a way that the new variance expressions presented here encompass pre existing ones as special cases via this latter analysis a new perspective emerges on recent work pertaining to the use of orthonormal basis structures in system identification namely that orthonormal bases are much more than an implementational option offering improved numerical properties in fact they are an intrinsic part of estimation since as shown here orthonormal bases quantify the asymptotic variability of the estimates whether or not they are actually employed in calculating them

brown s spectral distribution measure for r diagonal elements in finite von neumann algebras abstract in 1983 l g brown introduced a spectral distribution measure for non normal elements in a finite von neumann algebra m with respect to a fixed normal faithful tracial state in this paper we compute brown s spectral distribution measure in case t has a polar decomposition t uh where u is a haar unitary and u and h are free when ker t 0 this is equivalent to that t t is an r diagonal pair in the sense of nica and speicher the measure t is expressed explicitly in terms of the s transform of the distribution t t of the positive operator t t in case t is a circular element i e t x1 ix2 2 where x1 x2 is a free semicircular system then sp t d the closed unit disk and t has constant density 1 on d

induced ginibre ensemble of random matrices and quantum operations a generalization of the ginibre ensemble of non hermitian random square matrices is introduced the corresponding probability measure is induced by the ensemble of rectangular gaussian matrices via a quadratization procedure we derive the joint probability density of eigenvalues for such an induced ginibre ensemble and study various spectral correlation functions for complex and real matrices and analyse universal behaviour in the limit of large dimensions in this limit the eigenvalues of the induced ginibre ensemble cover uniformly a ring in the complex plane the real induced ginibre ensemble is shown to be useful to describe the statistical properties of evolution operators associated with random quantum operations for which the dimensions of the input state and the output state do differ

a unifying construction of orthonormal bases for system identification this paper develops a general and very simple construction for complete orthonormal bases for system identification this construction provides a unifying formulation of many previously studied orthonormal bases since the common fir and recently popular laguerre and two parameter kautz model structures are restrictive special cases of the construction presented here however in contrast to these special cases the basis vectors in the unifying construction of this paper can have arbitrary placement of pole position according to the prior information the user wishes to inject results characterizing the completeness of the bases and the accuracy properties of models estimated using the bases are provided

state space representation for arbitrary orthogonal rational functions abstract we first give a brief summary of the arguments that lead to the conclusion that the use of orthogonal rational functions in system identification should not be restricted to the takenaka malmquist basis because there are important cases where orthogonality with respect to a certain nonnegative weight function is required the main result however is a generalization of known state space descriptions for the takenaka malmquist basis to the more general case this is important for control problems

identification of rational approximate models in h sup spl infin using generalized orthonormal basis robust identification with fir models fails to be successful when the number of coefficients to be estimated becomes large caused by lightly damped modes of the plant or poles very close to the unit circle the paper presents a two stage algorithm to obtain a low order approximate model in frequency domain in a generalized orthonormal basis with guaranteed h sup spl infin error bound for deterministic linear time invariant stable systems the first stage being an l sup spl infin rational approximation and the second nonlinear step being an h sup spl infin rational approximation

support convergence in the single ring theorem we study 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 all these eigenvalues lie in a single ring

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