replica treatment of non hermitian disordered hamiltonians we employ the fermionic and bosonic replicated nonlinear models to treat ginibre unitary symplectic and orthogonal ensembles of non hermitian random matrix hamiltonians using the saddle point approach combined with the borel resummation procedure we derive the exact large n results for microscopic density of states in all three ensembles we also obtain tails of the density of states as well as the two point function for the unitary ensemble

fourier inversion theorem in mesoscopic transport abstract the fourier inversion theorem on the graded hyperbolic plane a supermanifold closely analogous to the quotient spaces of efetov s nonlinear models is described an application to the calculation of the average conductance of quasi one dimensional disordered samples is given

character expansion method for supergroups and extended superversions of the leutwyler smilga and berezin karpelevich integrals we introduce an extension of the character expansion method to the case of supergroups this method allows us to calculate a superversion of the leutwyler smilga integral which to the best of our knowledge has not been calculated before we also use the method to generalize a previously calculated superversion of the berezin karpelevich integral our character expansion method should also allow for the calculation of other supergroup integrals

negative moments of characteristic polynomials of random goe matrices and singularity dominated strong fluctuations we calculate the negative integer moments of the regularized characteristic polynomials of n n random matrices taken from the gaussian orthogonal ensemble goe in the limit as n the results agree nontrivially with a recent conjecture of berry and keating motivated by techniques developed in the theory of singularity dominated strong fluctuations this is the first example where nontrivial predictions obtained using these techniques have been proved

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

matrix pade approximation definitions and properties abstract some proposals are made to give a general definition of matrix pade approximants depending on the normalization of the denominator we define type i constant term is the unit matrix and type ii by conditions on the leading coefficient approximants existence and uniqueness are considered determinant expressions are given and relations among types i ii and left right approximants are considered

structured matrices and unconstrained rational interpolation problems we describe a fast recursive algorithm for the solution of an unconstrained rational interpolation problem by exploiting the displacement structure concept we use the interpolation data to implicitly define a convenient non hermitian structured matrix and then apply a computationally efficient procedure for its triangular factorization this leads to a transmission line interpretation that makes evident the interpolation properties we further discuss connections with the lagrange interpolating polynomial as well as questions regarding the minimality and the admissible degrees of complexity of the solutions

toward a theory of the integer quantum hall transition continuum limit of the chalker coddington model an n channel generalization of the network model of chalker and coddington is considered the model for n 1 is known to describe the critical behavior at the plateau transition in systems exhibiting the integer quantum hall effect using a recently discovered equality of integrals the network model is transformed into a lattice field theory defined over efetov s model space with unitary symmetry the transformation is exact for all n no saddle point approximation is made and no massive modes have to be eliminated the naive continuum limit of the lattice theory is shown to be a supersymmetric version of pruisken s nonlinear model with couplings xx n 4 and xy n 2 at the symmetric point it follows that the model for n 2 which describes a spin degenerate landau level and the random flux problem is noncritical on the basis of symmetry considerations and inspection of the hamiltonian limit a modified network model is formulated which still lies in the quantum hall universality class the prospec

on the efetov wegner terms by diagonalizing a hermitian supermatrix the diagonalization of hermitian supermatrices is studied such a change of coordinates is inevitable if one wishes to find certain structures in random matrix theory however it still poses serious problems since until now the calculation of all rothstein contributions known as efetov wegner terms in physics was quite cumbersome we derive the supermatrix bessel function with all efetov wegner terms for an arbitrary rotation invariant probability density function as applications we consider representations of generating functions for hermitian random matrices with and without an external field as integrals over eigenvalues of hermitian supermatrices all results are obtained with all efetov wegner terms which were previously unknown in such an explicit and compact representation

linear complexity parallel algorithms for linear systems of equations with recursive structure abstract a method of derivation of parallel algorithms for n 1 n 1 matrices with recursive structure is presented and applied to toeplitz hankel and other toeplitz like matrices the derived algorithms executed on o n parallel processors require o n arithmetic operations per processor

on the density of states of sparse random matrices the supersymmetric method of calculation of the density of states of a sparse random matrix is shown to be absolutely equivalent to the replica trick a functional generalization of the hubbard stratonovich transformation was used in the course of the solution

recursive construction for a class of radial functions ii superspace we extend the recursion formula for matrix bessel functions which we obtained previously to superspace it is sufficient to do this for the unitary orthosymplectic supergroup by direct computations we show that fairly explicit results can be obtained at least up to dimension 8 8 for the supermatrices since we introduce a new technique we discuss various of its aspects in some detail

derivation of determinantal structures for random matrix ensembles in a new way there are several methods to treat ensembles of random matrices in symmetric spaces circular matrices chiral matrices and others orthogonal polynomials and the supersymmetry method are particular powerful techniques here we present a new approach to calculate averages over ratios of characteristic polynomials at first sight paradoxically one can call our approach supersymmetry without supersymmetry because we use structures from supersymmetry without actually mapping onto superspaces we address two kinds of integrals which cover a wide range of applications for random matrix ensembles for probability densities factorizing in the eigenvalues we find determinantal structures in a unifying way as a new application we derive an expression for the k point correlation function of an arbitrary rotation invariant probability density over the hermitian matrices in the presence of an external field

a comparison of the superbosonization formula and the generalized hubbard stratonovich transformation recently two different approaches were put forward to extend the supersymmetry method in random matrix theory from gaussian ensembles to general rotation invariant ensembles these approaches are the generalized hubbard stratonovich transformation and the superbosonization formula here we prove the equivalence of both approaches to this end we reduce integrals over functions of supersymmetric wishart matrices to integrals over quadratic supermatrices of certain symmetries

asymptotic corrections to the wigner semicircular eigenvalue spectrum of a large real symmetric random matrix using the replica method the replica method has previously been used to calculate the semicircular averaged eigenvalue spectrum of the gaussian orthogonal ensemble of real symmetric n n random matrices in the limit where n to infinity the authors develop a perturbative scheme which within this same replica framework is used to calculate the corrections within this semicircular band of eigenvalues to order 1 n and 1 n2 a new and straightforward self consistency argument is presented and used to derive the shape of the averaged eigenvalue spectrum when n is large but finite and the scaling behaviour of this averaged eigenvalue spectrum near the band edges is demonstrated in a straightforward fashion some comments are made on the relation of the results to those of field theoretical calculations in zero dimensions

paried cauchy matrices abstract invertibility kernel description and inversion of some cauchy type matrices paired cauchy matrices are reduced to a rational interpolation problem the result is applied to some problems appearing in connection with the solution of nonlinear integrable equations furthermore connections between paired cauchy matrices and paired vandermonde and loewner matrices are presented

necessary conditions for constant solutions of coupled riccati equations in nash games abstract the asymptotic behaviour of solutions as t for coupled matrix riccati equations occurring in open loop linear quadratic nash games is studied in this paper a general formula representing all possible solutions is given necessary conditions for constant real solutions are derived and an estimate for the rate of convergence is obtained two examples illustrate these results

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

efficient solution of linear systems of equations with recursive structure abstract inversion algorithms of order n2 for n n matrices with recursive structure are derived in a unified way several known algorithms as well as some new ones for toeplitz type hilbert type and vandermonde type matrices are given

an inversion formula and fast algorithms for cauchy vandermonde matrices abstract matrices of composed type consisting of a vandermonde and a cauchy part and their connection to rational interpolation are investigated an inversion formula is presented and fast solution algorithms with o n log2 n complexity for cauchy vandermonde systems are developed

a new approach to the rational interpolation problem the vector case we generalize our earlier results on rational interpolation which were given in van barel and bultheel this journal 1990 for the scalar case and in bultheel and van barel 1990 for the vector case when all the interpolation points coincide to the case of vector data given at arbitrary points that may coincide or not this is the vector valued newton pade problem we give a recursive algorithm which has the important advantage over other algorithms that we do not need a reordering of the given interpolation data to overcome a singularity in the interpolation table not even in the nonnormal vector case it also generates all the information needed to give all solutions of the problem

optimum estimation via gradients of partition functions and information measures a statistical mechanical perspective in continuation to a recent work on the statistical mechanical analysis of minimum mean square error mmse estimation in gaussian noise via its relation to the mutual information the i mmse relation here we propose a simple and more direct relationship between optimum estimation and certain information measures e g the information density and the fisher information which can be viewed as partition functions and hence are amenable to analysis using statistical mechanical techniques the proposed approach has several advantages most notably its applicability to general sources and channels as opposed to the i mmse relation and its variants which hold only for certain classes of channels e g additive white gaussian noise channels we then demonstrate the derivation of the conditional mean estimator and the mmse in a few examples two of these examples turn out to be generalizable to a fairly wide class of sources and channels for this class the proposed approach is shown to yield an approximate conditional mean estimator and an mmse formula that has the flavor of a single letter expression we also show how our approach can easily be generalized to situations of mismatched estimation

critique of the replica trick it is shown that the replica trick fails to give the correct non perturbative result for the two point function s2 of the gaussian unitary ensemble of n n random matrices the failure arises from an incorrect description of the symmetries of the random matrix system in the limit n to infinity the correct description which involves integration over both non compact and compact degrees of freedom is obtained by using the method of superfields some implications for the localisation transition in disordered electronic systems and the theory of the quantised hall effect are suggested

random matrix theory by the supersymmetry method beyond the sigma model abstract the leading correction to the smoothed connected energy density density correlation function is obtained for the large energy difference within the context of the gaussian random matrix theory in order to achieve this result the supersymmetry method is extended beyond the sigma model to include small quadratic fluctuations around the saddle point special care is taken to avoid the potential divergence arising from the unbounded nature of the saddle point also in the small energy difference regime the leading correction to a two point correlation function is obtained

replica method for wide correlators in gaussian orthogonal unitary and symplectic random matrix ensembles we calculate connected correlators in gaussian orthogonal unitary and symplectic random matrix ensembles by the replica method in the 1 n expansion we obtain averaged one point green s functions up to the next to leading order wide two level correlators up to the first non trivial order and wide three level correlators up to the first non trivial order by carefully treating fluctuations in saddle point evaluation

a new formal approach to the rational interpolation problem an elegant and fast recursive algorithm is developed to solve the rational interpolation problem in a complementary way compared to existing methods we allow confluent interpolation points poles and infinity as one of the interpolation points not only one specific solution is given but a nice parametrization of all solutions we also give a linear algebra interpretation of the problem showing that our algorithm can also be used to handle a specific class of structured matrices

eigenvalue correlations in non hermitean symplectic random matrices a correlation function of complex eigenvalues of n n random matrices drawn from non hermitean random matrix ensembles of symplectic symmetry is given in terms of a quaternion determinant spectral properties of gaussian ensembles are studied in detail in the regimes of weak and strong non hermiticity

necessary and sufficient condition for two player stackelberg strategy this technical note revisits the open loop stackelberg strategy for a two player game by introducing a new costate which captures the future information of the control input we present a necessary and sufficient condition for the existence and uniqueness of the two player game the optimal strategy is designed in terms of three decoupled and symmetric riccati equations which improves the existing results greatly on computation

generalized confluent cauchy and cauchy vandermonde matrices abstract the so called generalized confluent cauchy vandermonde gccv matrices of the form c v consisting of a generalized confluent cauchy part c and a generalized confluent vandermonde part v are considered a simple relationship between gccv and classical confluent cauchy vandermonde ccv matrices is given this leads to the reduction of the displacement structure inversion formulas and factorizations of gccv matrices and the interpolation interpretations of linear systems with such gccv coefficient matrices as tangential interpolation problems to the corresponding results of ccv matrices the criteria of invertibility and left right inversion formulas for such matrices are given all results are stated for the general case of multiple interpolation nodes extending the work of heinig and vavrin among others

integrability of zero dimensional replica field theories at β 1 building on insights from the theory of integrable lattices the integrability is claimed for nonlinear replica sigma models derived in the context of real symmetric random matrices specifically the fermionic and the bosonic replica partition functions are proven to form a single supersymmetric pfaff kp hierarchy whose replica limit is shown to reproduce the celebrated nonperturbative formula for the density density eigenvalue correlation function in the infinite dimensional gaussian orthogonal ensemble implications of the formalism outlined are briefly discussed

an open loop stackelberg strategy for the linear quadratic mean field stochastic differential game this paper is concerned with the open loop linear quadratic lq stackelberg game of the mean field stochastic systems in finite horizon by means of two generalized differential riccati equations the follower first solves a mean field stochastic lq optimal control problem then the leader turns to solve an optimization problem for a linear mean field forward backward stochastic differential equation by introducing new state and costate variables we present a sufficient condition for the existence and uniqueness of the stackelberg strategy in terms of the solvability of some riccati equations and a convexity condition furthermore it is shown that the open loop stackelberg equilibrium admits a feedback representation involving the new state and its mean finally two examples are given to show the effectiveness of the proposed results

discrete time robust hierarchical linear quadratic dynamic games in this paper the theory of robust min max control is extended to hierarchical and multiplayer dynamic games for linear quadratic discrete time systems the proposed game model consists of one leader and many followers while the performance of all players is affected by disturbance the stackelberg nash saddle equilibrium point of the game is derived and a necessary and sufficient condition for the existence and uniqueness of such a solution is obtained in the infinite time horizon it is shown that the solution of the riccati equation is upper bounded under a condition that is called individual controllability by assuming such a condition and using a time varying lyapunov function the input to state stability of the hierarchical dynamic game is achieved considering the optimal feedback strategies of the players and an arbitrary disturbance as the input

infinite horizon linear quadratic stackelberg games for discrete time stochastic systems abstract in this paper we consider infinite horizon linear quadratic stackelberg games for a discrete time stochastic system with multiple decision makers necessary conditions for the existence of the stackelberg strategy set are derived in terms of the solvability of cross coupled stochastic algebraic equations csaes as an important application the hierarchical h constraint control problem for the infinite horizon discrete time stochastic system with multiple channel inputs is solved using the stackelberg game approach computational methods for solving the csaes are also discussed a numerical example is provided to demonstrate the usefulness of the proposed algorithms

displacement structure theory and applications in this survey paper we describe how strands of work that are important in two different fields matrix theory and complex function theory have come together in some work on fast computational algorithms for matrices with what we call displacement structure in particular a fast triangularization procedure can be developed for such matrices generalizing in a striking way an algorithm presented by schur 1917 j reine angew math 147 1917 pp 205 232 in a paper on checking when a power series is bounded in the unit disc this factorization algorithm has a surprisingly wide range of significant applications going far beyond numerical linear algebra we mention among others inverse scattering analytic and unconstrained rational interpolation theory digital filter design adaptive filtering and state space least squares estimation

continued fractions associated with the newton pade table we discuss first the block structure of the newton pade table or rational interpolation table corresponding to the double sequence of rational interpolants for the data zk h zk k 0 the m n entry of this table is the rational function of type m n solving the linearized rational interpolation problem on the firstm n 1 data we then construct continued fractions that are associated with either a diagonal or two adjacent diagonals of this newton pade table in such a way that the convergents of the continued fractions are equal to the distinct entries on this diagonal or this pair of diagonals respectively the resulting continued fractions are generalizations of thiele fractions and of magnus sp fractions a discussion of an some new results on related algorithms of werner and graves morris and hopkins are also given

wigner dyson statistics from the replica method we compute the correlation functions of the eigenvalues in the gaussian unitary ensemble using the fermionic replica method we show that non trivial saddle points which break replica symmetry must be included in the calculation in order to correctly reproduce the exact asymptotic behaviour of the correlation functions at a large distance

universal correlations of trapped one dimensional impenetrable bosons we calculate the asymptotic behaviour of the one body density matrix of one dimensional impenetrable bosons in finite size geometries our approach is based on a modification of the replica method from the theory of disordered systems we obtain explicit expressions for oscillating terms similar to fermionic friedel oscillations these terms are universal and originate from the strong short range correlations between bosons in one dimension

rational interpolation and the euclidean algorithm abstract a general framework leading to a parametrization of all rational functions which interpolate a given set of pairs of points is investigated this framework is based on the euclidean algorithm the resulting parametrization has the property that it keeps track of the complexity of both the numerator and of the denominator polynomials of the interpolating functions the cauchy interpolation problem and the related pade approximation problem can be treated within this framework

h sup spl infin control and estimation with preview part ii fixed size are solutions in discrete time h sup spl infin preview control and fixed lag smoothing problems are solved in general discrete time linear systems via a reduction to equivalent open loop differential games to prevent high order riccati equations found in some solutions the state of the hamilton jacobi system resides in a quotient space of an auxiliary extended state space system the dimension of that auxiliary space is equal to the state space dimension of the original system ignoring the delay

on linear quadratic stackelberg games with time preference rates this note deals with linear quadratic stackelberg differential games including time preference rates with an open loop information structure the properties of the characteristic matrix associated with the necessary conditions for a stackelberg strategy are pointed out it is shown that such a matrix exhibits a special symmetry property of its eigenvalues sufficient conditions to guarantee a predefined degree of stability are given based on the distribution of the eigenvalues in the complex plane

a new approach to derive pfaffian structures for random matrix ensembles correlation functions for matrix ensembles with orthogonal and unitary symplectic rotation symmetry are more complicated to calculate than in the unitary case the supersymmetry method and the orthogonal polynomials are two techniques to tackle this task recently we presented a new method to average ratios of characteristic polynomials over matrix ensembles invariant under the unitary group here we extend this approach to ensembles with orthogonal and unitary symplectic rotation symmetry we show that pfaffian structures can be derived for a wide class of orthogonal and unitary symplectic rotation invariant ensembles in a unifying way this also includes those for which this structure was not known previously as the real ginibre ensemble and the gaussian real chiral ensemble with two independent matrices as well