level dynamics in pseudointegrable billiards an experimental study abstract the level dynamics of pseudointegrable systems with different genus numbers g is studied experimentally using microwave cavities for higher energies the distribution of the eigenvalue velocities is gaussian as it is expected for chaotic systems with time reversal symmetry and shows no dependence on g also the curvature distribution p k for large k is decaying as it is expected for chaotic systems i e p k k 3 for small k an intermediate behavior is found where p k changes from integrable towards chaotic behavior with growing g

characterization of quantum chaos by the autocorrelation function of spectral determinants the autocorrelation function of spectral determinants is proposed as a convenient tool for the characterization of spectral statistics in general and for the study of the intimate link between quantum chaos and the random matrix theory in particular for this purpose the correlation functions of spectral determinants are evaluated for various random matrix ensembles and are compared with the corresponding semiclassical expressions the method is demonstrated by applying it to the spectra of the quantized sinai billiards in two and three dimensions

on the local theory of regular analytic matrix functions abstract a concise exposition is given of the local spectral theory of regular analytic matrix functions without any reference to elements from the global theory

paper zeros of sampled systems the zeros of the discrete time system obtained when sampling a continuous time system are explored theorems for the limiting zeros for large and small sampling periods are given conditions which guarantee that the sampled system has stable zeros are also presented

a contour integral approach to the computation of invariant pairs abstract we study some aspects of the invariant pair problem for matrix polynomials as introduced by betcke and kressner 3 and by beyn and thummler 6 invariant pairs extend the notion of eigenvalue eigenvector pairs providing a counterpart of invariant subspaces for the nonlinear case we compute formulations for the condition numbers and the backward error for invariant pairs and solvents we then adapt the sakurai sugiura moment method 1 to the computation of invariant pairs including some classes of problems that have multiple eigenvalues numerical refinement via a variant of newton s method is also studied furthermore we investigate the relation between the matrix solvent problem and the triangularization of matrix polynomials

scattering reflection and impedance of waves in chaotic and disordered systems with absorption we review recent progress in analysing wave scattering in systems with both intrinsic chaos and or disorder and internal losses when the scattering matrix is no longer unitary by mapping the problem onto a nonlinear supersymmetric model we are able to derive closed form analytic expressions for the distribution of reflection probability in a generic disordered system one of the most important properties resulting from such an analysis is statistical independence between the phase and the modulus of the reflection amplitude in every perfectly open channel the developed theory has far reaching consequences for many quantities of interest including local green functions and time delays in particular we point out the role played by absorption as a sensitive indicator of mechanisms behind the anderson localization transition we also provide a random matrix based analysis of s matrix and impedance correlations for various symmetry classes as well as the distribution of transmitted power for systems with broken time reversal invariance completing previous works on the subject the results can be applied in particular to the experimentally accessible impedance and reflection in a microwave or an ultrasonic cavity attached to a system of antennas

universal eigenvector correlations in quaternionic ginibre ensembles non hermitian random matrices enjoy non trivial correlations in the statistics of their eigenvectors we study the overlap among left and right eigenvectors in ginibre ensembles with quaternion val

a relation between billiard geometry and the temperature of its eigenvalue gas according to a conjecture of yukawa the parametric motion of the eigenvalues of a chaotic system leads to a phase space distribution proportional to where e is the energy of the eigenvalue gas and is its reciprocal temperature to test the conjecture in a first step correspondence between the well known pechukas yukawa level dynamics and that of a billiard with variable length is established next is expressed in terms of the billiard geometry thus fixing the only free parameter of the model finally experimental distributions of eigenvalue velocities curvatures etc obtained from sinai microwave billiards are analysed in terms of the model in all cases a quantitative agreement was found apart from some small deviations caused by the dominating bouncing ball orbit

on random matrix averages involving half integer powers of goe characteristic polynomials correlation functions involving products and ratios of half integer powers of characteristic polynomials of random matrices from the gaussian orthogonal ensemble goe frequently arise in applications of random matrix theory rmt to physics of quantum chaotic systems and beyond we provide an explicit evaluation of the large n limits of a few non trivial objects of that sort within a variant of the supersymmetry formalism and via a related but different method as one of the applications we derive the distribution of an off diagonal entry k ab of the resolvent or wigner k matrix of goe matrices which among other things is of relevance for experiments on chaotic wave scattering in electromagnetic resonators

statistics of resonances and delay times in random media beyond random matrix theory we review recent developments in quantum scattering from mesoscopic systems various spatial geometries whose closed analogues show diffusive localized or critical behaviour are considered these are the features that cannot be described by the universal random matrix theory results instead one has to go beyond this approximation and incorporate them in a non perturbative way here we pay particular attention to the traces of these non universal characteristics in the distribution of the wigner delay times and resonance widths the former quantity captures time dependent aspects of quantum scattering while the latter is associated with the poles of the scattering matrix

comments with reply on optimal rejection of persistent disturbances robust stability and mixed sensitivity minimization by m a dahleh and j b pearson the commenters point out that the sufficient condition for robust stability given in the above titled work see ibid vol 33 no 8 p 722 731 aug 1988 should be modified in their reply the authors agree with the commenters and describe the assumptions which led to the omission

choice of sampling to ensure minimum phase behaviour sufficient conditions for ensuring that a continuous stable proper minimum phase plant remains minimum phase when sampled have been derived as a corollary the critical sampling frequency the fastest sampling frequency ensuring minimum phase behaviour is shown to satisfy a simple equation an engineering method to obtain the critical sampling by use of a relay is proposed this method requires no prior knowledge of the plant

an exact formula for general spectral correlation function of random hermitian matrices we have found an exact formula expressing a general correlation function containing both products and ratios of characteristic polynomials of random hermitian matrices the answer is given in the form of a determinant an essential difference from the previously studied correlation functions of products only is the appearance of non polynomial functions along with the orthogonal polynomials these non polynomial functions are the cauchy transforms of the orthogonal polynomials the result is valid for arbitrary ensemble of 2 symmetry class and generalizes recent asymptotic formulae obtained for gaussian unitary ensemble and its chiral counterpart by different methods

exact distributions of eigenvalue curvatures for time reversal invariant chaotic systems the parametric motion of eigenvalues of chaotic quantum systems is studied by means of the distribution of eigenvalue curvatures k second derivative with respect to a perturbation parameter using supersymmetric integral representations this distribution is computed exactly for the orthogonal ensuremath beta 1 and the symplectic ensuremath beta 4 ensemble it is found that p k ensuremath propto 1 mathit k 2 mathrm ensuremath 2 mathrm ensuremath beta 2 in agreement with a recent conjecture by zakrzewski and delande phys rev e 47 1650 1993

first principles derivation of static avalanche size distributions we study the energy minimization problem for an elastic interface in a random potential plus a quadratic well as the position of the well is varied the ground state undergoes jumps called shocks or static avalanches we introduce an efficient and systematic method to compute the statistics of avalanche sizes and manifold displacements the tree level calculation i e mean field limit is obtained by solving a saddle point equation graphically it can be interpreted as the sum of all tree graphs the 1 loop corrections are computed using results from the functional renormalization group at the upper critical dimension the shock statistics is described by the brownian force model bfm the static version of the so called alessandro beatrice bertotti montorsi abbm model in the nonequilibrium context of depinning this model can itself be treated exactly in any dimension and its shock statistics is that of a levy process contact is made with classical results in probability theory on the burgers equation with brownian initial conditions in particular we obtain a functional extension of an evolution equation introduced by carraro and duchon which recursively constructs the tree diagrams in the field theory

an integral method for solving nonlinear eigenvalue problems we propose a numerical method for computing all eigenvalues and the corresponding eigenvectors of a nonlinear holomorphic eigenvalue problem that lie within a given contour in the complex plane the method uses complex integrals of the resolvent operator applied to at least k column vectors where k is the number of eigenvalues inside the contour the theorem of keldysh is employed to show that the original nonlinear eigenvalue problem reduces to a linear eigenvalue problem of dimension k no initial approximations of eigenvalues and eigenvectors are needed the method is particularly suitable for moderately large eigenvalue problems where k is much smaller than the matrix dimension we also give an extension of the method to the case where k is larger than the matrix dimension the quadrature errors caused by the trapezoid sum are discussed for the case of analytic closed contours using well known techniques it is shown that the error decays exponentially with an exponent given by the product of the number of quadrature points and the minimal distance of the eigenvalues to the contour

kronecker bases for linear matrix equations with application to two parameter eigenvalue problems abstract the general solutions of the homogeneous matrix equation axct bxdt 0 and the system of the matrix equations ax by 0 xct ydt 0 are described in terms of kronecker canonical forms i e in terms of kronecker invariants and kronecker bases for pairs of matrices a b and c d a canonical form for a pair of commuting matrices e f such that e2 f2 ef 0 is discussed these results are applied to construct a canonical basis for the second root subspace of a two parameter eigenvalue problem the corresponding relations for canonical invariants are given

s matrix poles for chaotic quantum systems as eigenvalues of complex symmetric random matrices from isolated to overlapping resonances we study complex eigenvalues of large symmetric random matrices of the form where both and are real symmetric is a random gaussian and is such that when when the model can be used to describe the universal statistics of s matrix poles resonances in the complex energy plane we derive the ensuing distribution of the resonance widths which generalizes the well known distribution to the case of overlapping resonances we also consider a different class of almost real matrices when is random and uncorrelated with

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

replica symmetry breaking condition exposed by random matrix calculation of landscape complexity we start with a rather detailed general discussion of recent results of the replica approach to statistical mechanics of a single classical particle placed in a random n 1 dimensional gaussian landscape and confined by a spherically symmetric potential suitably growing at infinity then we employ random matrix methods to calculate the density of stationary points as well as minima of the associated energy surface this is used to show that for a generic smooth concave confining potentials the condition of the zero temperature replica symmetry breaking coincides with one signaling that both mean total number of stationary points in the energy landscape and the mean number of minima are exponential in n for such systems the annealed complexity of minima vanishes cubically when approaching the critical confinement whereas the cumulative annealed complexity vanishes quadratically different behaviour reported in our earlier short communication fyodorov et al in jetp lett 85 261 2007 was due to non analyticity of the hard wall confinement potential finally for the simplest case of parabolic confinement we investigate how the complexity depends on the index of stationary points in particular we show that in the vicinity of critical confinement the saddle points with a positive annealed complexity must be close to minima as they must have a vanishing fraction of negative eigenvalues in the hessian

large deviations of extreme eigenvalues of random matrices we calculate analytically the probability of large deviations from its mean of the largest smallest eigenvalue of random matrices belonging to the gaussian orthogonal unitary and symplectic ensembles in particular we show that the probability that all the eigenvalues of an n ifmmode times else texttimes fi n random matrix are positive negative decreases for large n as ensuremath sim mathrm exp ensuremath ensuremath beta ensuremath theta 0 n 2 where the parameter ensuremath beta characterizes the ensemble and the exponent ensuremath theta 0 mathrm ln 3 4 0 274 text 653 dots is universal we also calculate exactly the average density of states in matrices whose eigenvalues are restricted to be larger than a fixed number ensuremath zeta thus generalizing the celebrated wigner semicircle law the density of states generically exhibits an inverse square root singularity at ensuremath zeta

random matrices and complexity of spin glasses cern y abstract we give an asymptotic evaluation of the complexity of spherical p spin spin glass models via random matrix theory this study enables us to obtain detailed infor mation about the bottom of the energy landscape including the absolute minimum the ground state the other local minima and describe an interesting layered structure of the low critical values for the hamiltonians of these models we also show that our ap proach allows us to compute the related tap complexity and extend the results known in the physics literature as an independent tool we prove a ldp for the k th largest eigenvalue of the goe extending the results of bdg01 how many critical values of given index and below a given level does a typical random morse function have on a high dimensional manifold our work addresses this question in a very special case we look at certain natural random gaussian functions on the n dimensional sphere known as p spin spherical spin glass models we cannot yet answer the question above about the typical number but we can study thoroughly the mean number which we show is exponentially large in n we introduce a new identity based on the classical kac rice formula relating random matrix theory and the problem of counting these critical values using this identity and tools from random matrix theory we give an asymptotic evaluation of the complexity of these spherical spin glass models the complexity mentioned here is defined as the mean number of critical points of given index whose value is below or above a given level this includes the important question of counting the mean number of local minima below a given level and in particular the question of finding the ground state energy the minimal value of the hamiltonian we show that this question is directly related to the study of the edge of the spectrum of the gaussian orthogonal ensemble goe the question of computing the complexity of mean field spin glass models has recently been thoroughly studied in the physics literature see for example clr03 and the refer ences therein mainly for a different measure of the complexity i e the mean number of solutions to the thouless anderson palmer equations or tap complexity our approach to the complexity enables us to recover known results in the physics literature about tap complexity to compute the ground state energy when p is even and to describe an interesting layered structure of the low energy levels of the hamiltonians of these models which might prove useful for the study of the metastability of langevin dynamics for these models in longer time scales than those studied in bdg01 the paper is organised as follows in section 2 we give our main results in section 3 we prove two main formulas theorem 2 1 and 2 2 relating random matrix theory specif ically the goe and spherical spin glasses these formulas are direct consequences of the kac rice formula we learned the version needed here in the book at07 for another modern account see aw09 the main new ingredient is the fact that for spherical spin glass models the hessian of the hamiltonian at a critical point conditioned on the value of the hamiltonian is a symmetric gaussian random matrix with independent entries up to symmetry plus a diagonal matrix this implies in particular that it is possible to

extreme statistics of complex random and quantum chaotic states complex random states have the statistical properties of the gaussian and circular unitary ensemble eigenstates of random matrix theory even though their components are correlated by the normalization constraint it is nevertheless possible to derive compact analytic formulas for their extreme values statistical properties for all dimensionalities the maximum intensity result slowly approaches the gumbel distribution even though the variables are bounded whereas the minimum intensity result rapidly approaches the weibull distribution since random matrix theory is conjectured to be applicable to chaotic quantum systems we calculate the extreme eigenfunction statistics for the standard map with parameters at which its classical map is fully chaotic the statistical behaviors are consistent with the finite n formulas

statistical theory of multimode random lasers we present a semiclassical laser theory for multimode lasing in optical resonators with overlapping modes nonlinear saturation and mode competition are characterized in terms of left and right eigenmodes of a non hermitian operator the theory is applied to wave chaotic cavities and weakly disordered random media in the limit of sufficiently strong pumping we find that the mean number of laser peaks increases with the 1 3 power of the pump strength

certified parallelotope continuation for one manifolds starting from an initial solution continuation methods efficiently produce a sequence of points on a manifold typically defined as the solution set of an underconstrained system of equations they have a wide range of applications ranging from curve plotting to polynomial root finding by homotopy however classical methods cannot guarantee that the returned points all belong to the same connected component of the manifold i e they may jump from one component to another trying to overcome this issue has given birth to several sophisticated heuristics on the one hand and to guaranteed methods based on rigorous computations on the other hand in this paper we introduce a new rigorous predictor corrector continuation method based on interval computations its novelty lies in the fact that it uses parallelotopes as defined in a goldsztejn and l granvilliers a new framework for sharp and efficient resolution of ncsp with manifolds of solutions constraints 15 2010 pp 190 212 to enclose consecuti

minimization of a large scale quadratic function subject to a spherical constraint an important problem in linear algebra and optimization is the trust region subproblem minimize a quadratic function subject to an ellipsoidal or spherical constraint this basic problem has several important large scale applications including seismic inversion and forcing convergence in optimization methods existing methods to solve the trust region subproblem require matrix factorizations which are not feasible in the large scale setting this paper presents an algorithm for solving the large scale trust region subproblem that requires a fixed size limited storage proportional to the order of the quadratic and that relies only on matrix vector products the algorithm recasts the trust region subproblem in terms of a parameterized eigenvalue problem and adjusts the parameter with a superlinearly convergent iteration to find the optimal solution from the eigenvector of the parameterized problem only the smallest eigenvalue and corresponding eigenvector of the parameterized problem needs to be computed the implicitly restarted lanczos method is well suited to this subproblem

edge effects in some perturbations of the gaussian unitary ensemble a bordering of gaussian unitary ensemble matrices is considered in which the bordered row consists of zero mean complex gaussians n 0 2 in 0 2 off the diagonal and the real gaussian n 2 on the diagonal we compute the explicit form of the eigenvalue probability function for such matrices as well as that for matrices obtained by repeating the bordering the correlations are in general determinantal and in the single bordering case the explicit form of the correlation kernel is computed in the large n limit it is shown that and or can be tuned to induce a separation of the largest eigenvalue this effect is shown to be controlled by a single parameter universal correlation kernel

an algorithm for computing jordan chains and inverting analytic matrix functions abstract we present an efficient algorithm for obtaining a canonical system of jordan chains for an n n regular analytic matrix function a that is singular at the origin for any analytic vector function b we show that each term in the laurent expansion of a 1 b may be obtained from the previous terms by solving an n d n d linear system where d is the order of the zero of det a at 0 the matrix representing this linear system contains a 0 as a principal submatrix which can be useful if a 0 is sparse the last several iterations can be eliminated if left jordan chains are computed in addition to right jordan chains the performance of the algorithm in floating point and exact rational arithmetic is reported for several test cases the method is shown to be forward stable in floating point arithmetic

exact solution for the kardar parisi zhang equation with flat initial conditions we provide the first exact calculation of the height distribution at arbitrary time t of the continuum kardar parisi zhang kpz growth equation in one dimension with flat initial conditions we use the mapping onto a directed polymer with one end fixed one free and the bethe ansatz for the replicated attractive boson model we obtain the generating function of the moments of the directed polymer partition sum as a fredholm pfaffian our formula valid for all times exhibits convergence of the free energy i e kpz height distribution to the gaussian orthogonal ensemble tracy widom distribution at large time

sensitivity of the eigenfunctions and the level curvature distribution in quantum billiards in searching for the manifestations of the sensitivity of the eigenfunctions in quantum billiards with dirichlet boundary conditions with respect to the boundary data the normal derivative we have performed instead various numerical tests for the robnik billiard a quadratic conformal map of the unit disk for 600 shape parameter values where we have looked at the sensitivity of the energy levels with respect to the shape parameter we show the energy level flow diagrams for three stretches of fifty consecutive odd eigenstates each with index 1000 to 2000 in particular we have calculated the unfolded and normalized level curvature distribution and found that it continuously changes from a delta distribution for the integrable case circle to a broad distribution in the classically ergodic regime for some shape parameters the agreement with the goe von oppen formula is very good whereas we have also cases where the deviation from goe is significant and of physical origin in the intermediate case of mixed classical dynamics we have a semiclassical formula in the spirit of the berry robnik 1984 surmise here the agreement with theory is not good partially due to the localization phenomena which are expected to disappear in the semiclassical limit we stress that even for classically ergodic systems there is no global universality for the curvature distribution not even in the semiclassical limit

level curvature distribution in a model of two uncoupled chaotic subsystems we study distributions of eigenvalue curvatures for a block diagonal random matrix perturbed by a full random matrix the most natural physical realization of this model is a quantum chaotic system with some inherent symmetry such that its energy levels form two independent subsequences subject to a generic perturbation which does not respect the symmetry we describe analytically a crossover in the form of a curvature distribution with a tunable parameter namely the ratio of intersubsystem intrasubsystem coupling strengths we find that the peak value of the curvature distribution is much more sensitive to the changes in this parameter than the power law tail behavior this observation may help to clarify some qualitative features of the curvature distributions observed experimentally in acoustic resonances of quartz blocks

universal hypotrochoidic law for random matrices with cyclic correlations the celebrated elliptic law describes the distribution of eigenvalues of random matrices with correlations between off diagonal pairs of elements having applications to a wide range of physical and biological systems here we investigate the generalization of this law to random matrices exhibiting higher order cyclic correlations between k tuples of matrix entries we show that the eigenvalue spectrum in this ensemble is bounded by a hypotrochoid curve with k fold rotational symmetry this hypotrochoid law applies to full matrices as well as sparse ones and thereby holds with remarkable universality we further extend our analysis to matrices and graphs with competing cycle motifs which are described more generally by polytrochoid spectral boundaries

random matrices close to hermitian or unitary overview of methods and results the paper discusses recent progress in understanding statistical properties of eigenvalues of weakly non hermitian and non unitary random matrices the first type of ensembles is of the form i with being a large random n n hermitian matrix with independent entries deformed by a certain anti hermitian n n matrix i satisfying in the limit of large dimension n the condition tr 2 n tr 2 here can be either a random or just a fixed given hermitian matrix ensembles of such a type with 0 emerge naturally when describing quantum scattering in systems with chaotic dynamics and serve to describe resonance statistics related models are used to mimic complex spectra of the dirac operator with chemical potential in the context of quantum chromodynamics ensembles of the second type arising naturally in scattering theory of discrete time systems are formed by n n matrices with complex entries such that i for 0 this coincides with the circular unitary ensemble and 0 i describes deviation from unitarity our result amounts to answering statistically the following old question given the singular values of a matrix describe the locus of its eigenvalues we systematically show that the obtained expressions for the correlation functions of complex eigenvalues describe a non trivial crossover from wigner dyson statistics of real unimodular eigenvalues typical of hermitian unitary matrices to ginibre statistics in the complex plane typical of ensembles with strong non hermiticity tr 2 tr 2 when n finally we discuss scarce results available on eigenvector statistics for weakly non hermitian random matrices

energy landscape of the finite size spherical three spin glass model we study the three spin spherical model with mean field interactions and gaussian random couplings for moderate system sizes of up to 20 spins we obtain all stationary points of the energy landscape by means of the numerical polynomial homotopy continuation method on the basis of these stationary points we analyze the complexity and other quantities related to the glass transition of the model and compare these finite system quantities to their exact counterparts in the thermodynamic limit

minimizing a quadratic over a sphere a new method the sequential subspace method ssm is developed for the problem of minimizing a quadratic over a sphere in our scheme the quadratic is minimized over a subspace which is adjusted in successive iterations to ensure convergence to an optimum when a sequential quadratic programming iterate is included in the subspace convergence is locally quadratic numerical comparisons with other recent methods are given

aging of spherical spin glasses sompolinski and zippelius 1981 propose the study of dynamical systems whose invariant measures are the gibbs measures for hard to analyze statistical physics models of interest in the course of doing so physicists often report of an aging phenomenon for example aging is expected to happen for the sherrington kirkpatrick model a disordered mean field model with a very complex phase transition in equilibrium at low temperature we shall study the langevin dynamics for a simplified spherical version of this model the induced rotational symmetry of the spherical model reduces the dynamics in question to an n dimensional coupled system of ornstein uhlenbeck processes whose random drift parameters are the eigenvalues of certain random matrices we obtain the limiting dynamics for n approaching infinity and by analyzing its long time behavior explain what is aging mathematically speaking what causes this phenomenon and what is its relationship with the phase transition of the corresponding equilibrium invariant measures

a constrained eigenvalue problem in this paper we consider the following mathematical and computational problem given the quantities a n m by n m matrix symmetric n 0 n n m by m matrix with full rank t vector of dimension m with nt t 1 determine an x such that operatorname x t ax min subject to the constraints n t x t i x t x 1 ii variants of this problem occur in many applications 1 5 7 8 11 the problem has been studied previously when t 0 the null vector cf 4 6 when t 0 then the problem becomes more complicated

continuation of eigenvalues and invariant pairs for parameterized nonlinear eigenvalue problems invariant pairs have been proposed as a numerically robust means to represent and compute several eigenvalues along with the corresponding generalized eigenvectors for matrix eigenvalue problems that are nonlinear in the eigenvalue parameter in this work we consider nonlinear eigenvalue problems that depend on an additional parameter and our interest is to track several eigenvalues as this parameter varies based on the concept of invariant pairs a theoretically sound and reliable numerical continuation procedure is developed particular attention is paid to the situation when the procedure approaches a singularity that is when eigenvalues included in the invariant pair collide with other eigenvalues for the real generic case it is proven that such a singularity only occurs when two eigenvalues collide on the real axis it is shown how this situation can be handled numerically by an appropriate expansion of the invariant pair the viability of our continuation procedure is illustrated by a numerical example

distribution of the spacing between two adjacent avoided crossings we consider the frequency at which avoided crossings appear in an energy level structure when an external field is applied to a quantum chaotic system the distribution of the spacing in the parameter between two adjacent avoided crossings is investigated using a random matrix model we find that the distribution of these spacings is well fitted by a power law distribution for small spacings the powers are 2 and 3 for the gaussian orthogonal ensemble and gaussian unitary ensemble respectively we also find that the distributions decay exponentially for large spacings the distributions in concrete quantum chaotic systems agree with those of the random matrix model

reviving the method of particular solutions fox henrici and moler made famous a method of particular solutions for computing eigenvalues and eigenmodes of the laplacian in planar regions such as polygons we explain why their formulation of this method breaks down when applied to regions that are insufficiently simple and propose a modification that avoids these difficulties the crucial changes are to introduce points in the interior of the region as well as on the boundary and to minimize a subspace angle rather than just a singular value or a determinant similar methods may be used to improve other mesh free algorithms for a variety of computational problems

quantum limit of the laser line width in chaotic cavities and statistics of residues of scattering matrix poles he quantum limited line width of a laser cavity is enhanced above the schawlow townes value by the petermann factor k due to the non orthogonality of the cavity modes we derive the relation between the petermann factor and the residues of poles of the scattering matrix and investigate the statistical properties of the petermann factor for cavities in which the radiation is scattered chaotically for a single scattering channel we determine the complete probability distribution of k and find that the average petermann factor depends non analytically on the area of the opening and greatly exceeds the most probable value for an arbitrary number n of scattering channels we calculate as a function of the decay rate i of the lasing mode we find for n 1 that for typical values of i the average petermann factor propto sqrt n 1 is parametrically larger than unity

on statistics of bi orthogonal eigenvectors in real and complex ginibre ensembles combining partial schur decomposition with supersymmetry we suggest a method of studying the joint probability density jpd of an eigenvalue and the associated non orthogonality overlap factor also known as the eigenvalue condition number of the left and right eigenvectors for non selfadjoint gaussian random matrices of size n times n first we derive the general finite n expression for the jpd of a real eigenvalue lambda and the associated non orthogonality factor in the real ginibre ensemble and then analyze its bulk and edge scaling limits the ensuing distribution is maximally heavy tailed so that all integer moments beyond normalization are divergent a similar calculation for a complex eigenvalue z and the associated non orthogonality factor in the complex ginibre ensemble is presented as well and yields a distribution with the finite first moment its bulk scaling limit yields a distribution whose first moment reproduces the well known result of chalker and mehlig phys rev lett 81 16 3367 3370 1998 and we provide the edge scaling distribution for this case as well our method involves evaluating the ensemble average of products and ratios of integer and half integer powers of characteristic polynomials for ginibre matrices which we perform in the framework of a supersymmetry approach our paper complements recent studies by bourgade and dubach the distribution of overlaps between eigenvectors of ginibre matrices 2018 arxiv 1801 01219

continuation of invariant subspaces in large bifurcation problems we summarize an algorithm for computing a smooth orthonormal basis for an invariant subspace of a parameter dependent matrix and describe how to extend it for numerical bifurcation analysis we adapt the continued subspace to track behavior relevant to bifurcations and use projection methods to deal with large problems to test our ideas we have integrated our code into matcont a program for numerical continuation and bifurcation analysis

topology trivialization and large deviations for the minimum in the simplest random optimization finding the global minimum of a cost function given by the sum of a quadratic and a linear form in n real variables over n 1 dimensional sphere is one of the simplest yet paradigmatic problems in optimization theory known as the trust region subproblem or constraint least square problem when both terms in the cost function are random this amounts to studying the ground state energy of the simplest spherical spin glass in a random magnetic field we first identify and study two distinct large n scaling regimes in which the linear term magnetic field leads to a gradual topology trivialization i e reduction in the total number mathcal n tot of critical stationary points in the cost function landscape in the first regime mathcal n tot remains of the order n and the cost function energy has generically two almost degenerate minima with the tracy widom tw statistics in the second regime the number of critical points is of the order of unity with a finite probability for a single minimum in that case the mean total number of extrema minima and maxima of the cost function is given by the laplace transform of the tw density and the distribution of the global minimum energy is expected to take a universal scaling form generalizing the tw law though the full form of that distribution is not yet known to us one of its far tails can be inferred from the large deviation theory for the global minimum in the rest of the paper we show how to use the replica method to obtain the probability density of the minimum energy in the large deviation approximation by finding both the rate function and the leading pre exponential factor

squared eigenvalue condition numbers and eigenvector correlations from the single ring theorem we extend the so called single ring theorem feinberg and zee 1997 nucl phys b 504 579 also known as the haagerup larsen theorem haagerup and larsen 2000 j funct anal 176 331 we do this by showing that in the limit when the size of the matrix goes to infinity a particular correlator between left and right eigenvectors of the relevant non hermitian matrix x being the spectral density weighted by the squared eigenvalue condition number is given by a simple formula involving only the radial spectral cumulative distribution function of x we show that this object allows the calculation of the conditional expectation of the squared eigenvalue condition number we give examples and provide a cross check of the analytic prediction by the large scale numerics

large deviations of the maximum eigenvalue for wishart and gaussian random matrices we present a coulomb gas method to calculate analytically the probability of rare events where the maximum eigenvalue of a random matrix is much larger than its typical value the large deviation function that characterizes this probability is computed explicitly for wishart and gaussian ensembles the method is general and applies to other related problems e g the joint large deviation function for large fluctuations of top eigenvalues our results are relevant to widely employed data compression techniques namely the principal components analysis analytical predictions are verified by extensive numerical simulations

some special cases in the stability analysis of multi dimensional time delay systems using the matrix lambert w function this paper revisits a recently developed methodology based on the matrix lambert w function for the stability analysis of linear time invariant time delay systems by studying a particular yet common second order system we show that in general there is no one to one correspondence between the branches of the matrix lambert w function and the characteristic roots of the system furthermore it is shown that under mild conditions only two branches suffice to find the complete spectrum of the system and that the principal branch can be used to find several roots and not the dominant root only as stated in previous works the results are first presented analytically and then verified by numerical experiments

semiclassical quantization of billiards with mixed boundary conditions the semiclassical theory for billiards with mixed boundary conditions is developed and explicit expressions for the smooth and the oscillatory parts of the spectral density are derived the parametric dependence of the spectrum on the boundary condition is shown to be a very useful diagnostic tool in the semiclassical analysis of the spectrum of billiards it is also used to check in detail some recently proposed parametric spectral statistics the methods are illustrated in the analysis of the spectrum of the sinai billiard and its parametric dependence on the boundary condition on the dispersing arc