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

how many eigenvalues of a random matrix are real let a be an n x n matrix whose elements are independent random variables with standard normal distributions as n oo the expected number of real eigenvalues is asymptotic to v 7r we obtain a closed form expres sion for the expected number of real eigenvalues for finite n and a formula for the density of a real eigenvalue for finite n asymptotically a real normalized eigenvalue al ii of such a random matrix is uniformly distributed on the in terval 1 1 analogous but strikingly different results are presented for the real generalized eigenvalues we report on numerical experiments confirming these results and suggesting that the assumption of normality is not important for the asymptotic results department of mathematics massachusetts institute of technology cambridge massachusetts 02139 e mail address edelmanfmath mit edu arts and sciences kapi olani community college 4303 diamond head road honolulu hawaii 98616 e mail address kostlanfkahuna math hawaii edu tj watson rsearch center 32 2 ibm yorktown heights new york 10598 0218 e mail address shubqwatson ibm com this content downloaded from 157 55 39 29 on tue 12 apr 2016 08 54 01 utc all use subject to http about jstor org terms

transition from localized to extended eigenstates in the ensemble of power law random banded matrices we study statistical properties of the ensemble of large n times n random matrices whose entries h ij decrease in a power law fashion h ij sim i j alpha mapping the problem onto a nonlinear sigma model with non local interaction we find a transition from localized to extended states at alpha 1 at this critical value of alpha the system exhibits multifractality and spectral statistics intermediate between the wigner dyson and poisson one these features are reminiscent of those typical for the mobility edge of disordered conductors we find a continuous set of critical theories at alpha 1 parametrized by the value of the coupling constant of the sigma model at alpha 1 all states are expected to be localized with integrable power law tails at the same time for 1 alpha 3 2 the wave packet spreading at short time scale is superdiffusive langle r rangle sim t frac 1 2 alpha 1 which leads to a modification of the altshuler shklovskii behavior of the spectral correlation function at 1 2 alpha 1 the statistical properties of eigenstates are similar to those in a metallic sample in d alpha 1 2 1 dimensions finally the region alpha 1 2 is equivalent to the corresponding gaussian ensemble of random matrices alpha 0 the theoretical predictions are compared with results of numerical simulations

quantum graphs a simple model for chaotic scattering we connect quantum graphs with infinite leads and turn them into scattering systems we show that they display all the features which characterize quantum scattering systems with an underlying classical chaotic dynamics typical poles delay time and conductance distributions ericson fluctuations and when considered statistically the ensemble of scattering matrices reproduces quite well the predictions of the appropriately defined random matrix ensembles the underlying classical dynamics can be defined and it provides important parameters which are needed for the quantum theory in particular we derive exact expressions for the scattering matrix and an exact trace formula for the density of resonances in terms of classical orbits analogous to the semiclassical theory of chaotic scattering we use this in order to investigate the origin of the connection between random matrix theory and the underlying classical chaotic dynamics being an exact theory and due to its relative simplicity it offers new insights into this problem which is at the forefront of the research in chaotic scattering and related fields

correlations between eigenvalues of large random matrices with independent entries we derive the connected correlation functions for eigenvalues of large hermitian random matrices with independently distributed elements using both a diagrammatic and a renormalization group rg inspired approach with the diagrammatic method we obtain a general form for the one two and three point connected green functions for this class of ensembles when matrix elements are identically distributed and then discuss the derivation of higher order functions by the same approach using the rg approach we rederive the one and two point green functions and show they are unchanged by choosing certain ensembles with nonidentically distributed elements throughout we compare the green functions we obtain to those from the class of ensembles with unitary invariant distributions and discuss universality in both ensemble classes textcopyright 1996 the american physical society

eigenvalue statistics of random real matrices completing ginibre s work we determine the joint probability density of eigenvalues in a gaussian ensemble of real asymmetric matrices which is invariant under orthogonal transformations the symmetry parameter r may vary from 1 antisymmetric ensemble through 0 completely asymmetric ensemble to 1 symmetric ensemble the elliptic law for the average density of eigenvalues in the limit of large dimension is recovered matrices of the type considered appear in models for neural network dynamics and dissipative quantum dynamics

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

effective coupling for open billiards we derive an explicit expression for the coupling constants of individual eigenstates of a closed billiard that is opened by attaching a waveguide the wigner time delay and the resonance positions resulting from the coupling constants are compared to an exact numerical calculation deviations can be attributed to evanescent modes in the waveguide and to the finite number of eigenstates taken into account the influence of the shape of the billiard and of the boundary conditions at the mouth of the waveguide are also discussed finally we show that the mean value of the dimensionless coupling constants tends to the critical value when the eigenstates of the billiard follow random matrix theory

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

localization problem in one dimension mapping and escape a one dimensional schr odinger equation in a discontinuous quasiperiodic potential is reduced to a recursion relation for transfer matrices and then to one for traces of these matrices when the potential is periodic the bandwidth goes to zero as an algebraic function of the period with a critical index which depends upon the potential strength this critical index is also evaluated as the solution to an escape rate problem for the recursion relations

stable equilibria to a singularly perturbed reaction diffusion equation in a degenerated heterogeneous environment abstract we address the problem u t 2 u f u x in r n n 1 under boundary condition u 0 where f u x u a x u x u b x x a x b x 2 and a b in the novelty here lies in the fact that the roots of f are allowed to degenerate in the sense that a b in d where d is such that d d 1 d 2 d 1 d 2 d 1 and d 2 are non empty open connected sets with lipschitz continuous boundaries and a b in d for small we prove existence of four families of stable stationary solutions u approaching the roots of f in the topology of l 1 our approach is variational and based on convergence theory

secular determinants of random unitary matrices we consider the characteristic polynomials of random unitary matrices u drawn from various circular ensembles in particular the statistics of the coefficients of these polynomials are studied the variances of these secular coefficients are given explicitly for arbitrary dimension and continued analytically to arbitrary values of the level repulsion exponent the latter secular coefficients are related to the traces of powers of u by newton s well known formulae while the traces tend to have gaussian distributions and to be statistically independent among one another in the limit as the matrix dimension grows large the secular coefficients exhibit strong mutual correlations due to newton s mixing of traces to coefficients these results might become relevant for current efforts at combining semiclassics and random matrix theory in quantum treatments of classically chaotic dynamics

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

spectrum of the fokker planck operator representing diffusion in a random velocity field we study spectral properties of the fokker planck operator that represents particles moving via a combination of diffusion and advection in a time independent random velocity field presenting in detail work outlined elsewhere j t chalker and z j wang phys rev lett 79 1797 1997 we calculate analytically the ensemble averaged one particle green function and the eigenvalue density for this fokker planck operator using a diagrammatic expansion developed for resolvents of non hermitian random operators together with a mean field approximation the self consistent born approximation which is well controlled in the weak disorder regime for dimension dg2 the eigenvalue density in the complex plane is nonzero within a wedge that encloses the negative real axis particle motion is diffusive at long times but for short times we find a novel time dependence of the mean square displacement r 2 ensuremath sim t 2 d in dimension dg2 associated with the imaginary parts of eigenvalues

nonuniversal correlations for random matrix ensembles the eigenvalue correlations of the generalized gaussian and laguerre random matrix ensembles are calculated exactly the fluctuations are shown to be nonuniversal in certain intervals of the spectrum a physical example from quantum transport where such nonuniversal effects occur is discussed

random tensor theory extending random matrix theory to mixtures of random product states we consider a problem in random matrix theory that is inspired by quantum information theory determining the largest eigenvalue of a sum of p random product states in mathbb c d otimes k where k and p d k are fixed while d when k 1 the marcenko pastur law determines up to small corrections not only the largest eigenvalue 1 sqrt p d k 2 but the smallest eigenvalue min 0 1 sqrt p d k 2 and the spectral density in between we use the method of moments to show that for k 1 the largest eigenvalue is still approximately 1 sqrt p d k 2 and the spectral density approaches that of the marcenko pastur law generalizing the random matrix theory result to the random tensor case our bound on the largest eigenvalue has implications both for sampling from a particular heavy tailed distribution and for a recently proposed quantum data hiding and correlation locking scheme due to leung and winter

classical wave experiments on chaotic scattering we review recent research on the transport properties of classical waves through chaotic systems with special emphasis on microwaves and sound waves inasmuch as these experiments use antennas or transducers to couple waves into or out of the systems scattering theory has to be applied for a quantitative interpretation of the measurements most experiments concentrate on tests of predictions from random matrix theory and the random plane wave approximation in all studied examples a quantitative agreement between experiment and theory is achieved to this end it is necessary however to take absorption and imperfect coupling into account concepts that were ignored in most previous theoretical investigations classical phase space signatures of scattering are being examined in a small number of experiments

statistics of resonances and nonorthogonal eigenfunctions in a model for single channel chaotic scattering we describe analytical and numerical results on the statistical properties of complex eigenvalues and the corresponding nonorthogonal eigenvectors for non hermitian random matrices modeling one channel quantum chaotic scattering in systems with broken time reversal invariance

summing free unitary random matrices i use quaternion free probability calculus an extension of free probability to non hermitian matrices which is introduced in a succinct but self contained way to derive in the large size limit the mean densities of the eigenvalues and singular values of sums of independent unitary random matrices weighted by complex numbers in the case of circular unitary ensemble summands i write them in terms of two master equations which i then solve and numerically test in four specific cases i conjecture a finite size extension of these results exploiting the complementary error function i prove a central limit theorem and its first subleading correction for independent identically distributed zero drift unitary random matrices

the asymptotics of quantum max flow min cut the quantum max flow min cut conjecture relates the rank of a tensor network to the minimum cut in the case that all tensors in the network are identical in calegari et al j am math soc 23 1 107 188 2010 this conjecture was shown to be false in cui et al j math phys 57 062206 2016 by an explicit counter example here we show that the conjecture is almost true in that the ratio of the quantum max flow to the quantum min cut converges to 1 as the dimension n of the degrees of freedom on the edges of the network tends to infinity the proof is based on estimating moments of the singular values of the network we introduce a generalization of rainbow diagrams to tensor networks to estimate the dominant diagrams a direct comparison of second and fourth moments lower bounds the ratio of the quantum max flow to the quantum min cut by a constant to show the tighter bound that the ratio tends to 1 we consider higher moments in addition we show that the limiting moments as n agree with that in a different ensemble where tensors in the network are chosen independently this is used to show that the distributions of singular values in the two different ensembles weakly converge to the same limiting distribution we present also a numerical study of one particular tensor network which shows a surprising dependence of the rank deficit on n mod 4 and suggests further conjecture on the limiting behavior of the rank

correlation functions in disordered systems recently we found that the correlation between the eigenvalues of random hermitian matrices exhibits universal behavior here we study this universal behavior and develop a diagrammatic approach which enables us to extend our previous work to the case in which the random matrix evolves in time or varies as some external parameters vary we compute the current current correlation function discuss various generalizations and compare our work with the work of other authors we study the distribution of eigenvalues of hamiltonians consisting of a sum of a deterministic term and a random term the correlation between the eigenvalues when the deterministic term is varied is calculated

correlation functions of eigenvalues of multi matrix models and the limit of a time dependent matrix the universality of correlation functions of eigenvalues of large random matrices has been observed in various physical systems and proved in some particular cases as the hermitian one matrix model with polynomial potential here we consider the more difficult case of a unidimensional chain of hermitian matrices with first neighbour couplings and polynomial potentials an asymptotic expression of the orthogonal polynomials and a generalization of the darboux christoffel theorem allow us to find new results for the correlations of eigenvalues of different matrices of the chain eventually we consider the limit of the infinite chain of matrices which can be interpreted as a time dependent hermitian one matrix model and give the correlation functions of eigenvalues at different times

renormalizing rectangles and other topics in random matrix theory we consider random hermitian matrices made of complex or realm n rectangular blocks where the blocks are drawn from various ensembles these matrices haven pairs of opposite real nonvanishing eigenvalues as well asm n zero eigenvalues form n these zero eigenvalues are kinematical in the sense that they are independent of randomness we study the eigenvalue distribution of these matrices to leading order in the large n m limit in which the rectangularity r m n is held fixed we apply a variety of methods in our study we study gaussian ensembles by a simple diagrammatic method by the dyson gas approach and by a generalization of the kazakov method these methods make use of the invariance of such ensembles under the action of symmetry groups the more complicated wigner ensemble which does not enjoy such symmetry properties is studied by large n renormalization techniques in addition to the kinematical function spike in the eigenvalue density which corresponds to zero eigenvalues we find for both types of ensembles that if r 1 is held fixed asn then nonzero eigenvalues give rise to two separated lobes that are located symmetrically with respect to the origin this separation arises because the nonzero eigenvalues are repelled macroscopically from the origin finally we study the oscillatory behavior of the eigenvalue distribution near the endpoints of the lobes a behavior governed by airy functions asr 1 the lobes come closer and the airy oscillatory behavior near the endpoints that are close to zero breaks down we interpret this breakdown as a signal thatr 1 drives a crossover to the oscillation governed by bessel functions near the origin for matrices made of square blocks

universal behavior of correlations between eigenvalues of random matrices the universal connected correlations proposed recently between eigenvalues of unitary random matrices is examined numerically we perform an ensemble average by the monte carlo sampling although density of eigenvalues and a bare correlation of the eigenvalues are not universal the connected correlation shows a universal behavior after smoothing

non hermitian random matrix theory summation of planar diagrams the single ring theorem and the disc annulus phase transition i review aspects of work done in collaboration with a zee and r scalettar 1997 nucl phys b 504 579 1997 nucl phys b 501 643 2001 j math phys 42 5718 on complex non hermitian random matrices i open by explaining why the bag of tools used regularly in analysing hermitian random matrices cannot be applied directly to analyse non hermitian matrices and then introduce the method of hermitization which solves this problem then for rotationally invariant ensembles i derive a master equation for the average density of eigenvalues in the complex plane in the limit of infinitely large matrices this is achieved by resumming all the planar diagrams which appear in the perturbative expansion of the hermitized green function remarkably this resummation can be carried out explicitly for any rotationally invariant ensemble i prove that in the limit of infinitely large matrices the shape of the eigenvalue distribution is either a disc or an annulus this is the celebrated single ring theorem which of these shapes is realized is determined by the parameters coupling constants which determine the ensemble by varying these parameters a phase transition may occur between the two possible shapes i briefly discuss the universal features of this transition as the analysis of this problem relies heavily on summation of planar feynman diagrams i make a special effort at presenting a pedagogical exposition of the diagrammatic method which some readers may find useful

random transition rate matrices for the master equation random matrix theory is applied to transition rate matrices in the pauli master equation we study the distribution and correlations of eigenvalues which govern the dynamics of complex stochastic systems both the cases of identical and of independent rates of forward and backward transitions are considered the first case leads to symmetric transition rate matrices whereas the second corresponds to general asymmetric matrices the resulting matrix ensembles are different from the standard ensembles and show different eigenvalue distributions for example the fraction of real eigenvalues scales anomalously with matrix dimension in the asymmetric case

laguerre ensembles of random matrices nonuniversal correlation functions the eigenvalue correlation functions of the laguerre ensembles exponential ensembles of real symmetric and self dual quaternion random matrices are calculated it is shown that in certain intervals of the spectrum the renormalized correlations are not universal a physical example in the transmission spectrum of a disordered conductor where such nonuniversality occurs is given

large rectangular random matrices models with a multiplet of field variables arranged into rectangular matrices in the limit of infinite dimensions of the matrices are studied in zero dimensional space where the problem is a combinatorial one a closed solution is given that improves the one previously known in arbitrary space dimension a symmetry is described that connects rectangular models with vector models

statistical properties of chaotic scattering with one open channel the correspondence between statistical properties of decaying states and fluctuations in resonance scattering is studied in a statistical model with one open channel the model is described by an ensemble of random non hermitian matrices the dependence of the correlation length on the coupling parameter both for the s matrix and the cross section is studied numerically we show that maximal correlations in the scattering arise for a certain value of the coupling to the continuum reflecting a specific change in the internal motion of intermediate decaying systems also the fourier transform of the two point correlation function of the s matrix is analyzed both analytically and numerically the self averaging nature of this function is explicitly demonstrated

optical structures with local pt symmetry we propose a new class of optical synthetic materials that are described by non hermitian hamiltonians the building blocks of such systems are coupled symmetric elements dimers with coupling t despite the lack of global symmetry these systems have a robust parameter region of real spectra exact phase even in cases where the complex refractive index n i of each dimer is random the validity of our proposition is confirmed for representative cases where we calculate the borders of the exact phase in terms of and t

the effect of a singular perturbation on nonconvex variational problems we study the effect of a singular perturbation on certain nonconvex variational problems the goal is to characterize the limit of minimizers as some perturbation parameter 0 the technique utilizes the notion of convergence of variational problems developed by de giorgi the essential idea is to identify the first nontrivial term in an asymptotic expansion for the energy of the perturbed problem in so doing one characterizes the limit of minimizers as the solution of a new variational problem for the cases considered here these new problems have a simple geometric nature involving minimal surfaces and geodesics

local minimisers and singular perturbations we construct local minimisers to certain variational problems the method is quite general and relies on the theory of convergence the approach is demonstrated through the model problem it is shown that in certain nonconvex domains n and for e small there exist nonconstant local minimisers u e satisfying u e 1 except in a thin transition layer the location of the layer is determined through the requirement that in the limit u e u 0 the hypersurface separating the states u 0 1 and u 0 1 locally minimises surface area generalisations are discussed with for example vector valued u and anisotropic perturbations replacing u 2

computations of eisenstein series on fuchsian groups we present numerical investigations of the value distribution and distribution of fourier coefficients of the eisenstein series e z s on arithmetic and non arithmetic fuchsian groups our numer

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

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

spectral properties of one dimensional quasi crystals in this paper we prove that the one dimensional schrodinger operator onl 2 with potential given by upsilon n lambda chi 1 alpha 1 x n alpha alpha notin mathbb q has a cantor spectrum of zero lebesgue measure for any irrational and any 0 we can thus extend the kotani result on the absence of absolutely continuous spectrum for this model to all mathtype mtef 2 1 feaafiart1ev1aaatcvaufebsjuyzl2yd9gzlbvynv2caerbulwbln hiov2dgi1btfmbaexatlxbi9gbaerbd9wdylwzybitldharqqtubsr 4rnchbgeagqivu0je9sqqrpepc0xbbl8f4rqqrffpeea0xe9lq jc9 vqaqpepm0xbba9pwe9q8fs0 yqaqpepae9pg0firpepge9fr xfr x frpewzqaaeaabigaciaacaqabeaadaqaaqaaaoqaaiaadihaiiaacq wfiiiztuudjxwak1uy0hmmaehbfv3yslgzg0uy0hgiud3bagqbbiab nj8ubaa 4628 x in mathbb t

reducing nonideal to ideal coupling in random matrix description of chaotic scattering application to the time delay problem we write explicitly a transformation of the scattering phases reducing the problem of quantum chaotic scattering for systems with m statistically equivalent channels at nonideal coupling to that for ideal coupling unfolding the phases by their local density leads to universality of their local fluctuations for large m a relation between the partial time delays and diagonal matrix elements of the wigner smith matrix is revealed for ideal coupling this helped us in deriving the joint probability distribution of partial time delays and the distribution of the wigner time delay

almost hermitian random matrices crossover from wigner dyson to ginibre eigenvalue statistics by using the method of orthogonal polynomials we analyze the statistical properties of complex eigenvalues of random matrices describing a crossover from hermitian matrices characterized by the wigner dyson statistics of real eigenvalues to strongly non hermitian ones whose complex eigenvalues were studied by ginibre two point statistical measures as e g spectral form factor number variance and small distance behavior of the nearest neighbor distance distribution p s are studied in more detail in particular we found that the latter function may exhibit unusual behavior p s ensuremath propto s 5 2 for some parameter values

non hermitian random matrix models free random variable approach using the standard concepts of free random variables we show that for a large class of non hermitian random matrix models the support of the eigenvalue distribution follows from their hermitian analogs using a conformal transformation we also extend the concepts of free random variables to the class of non hermitian matrices and apply them to the models discussed by ginibre girko elliptic ensemble j ginibre j math phys bold 6 1440 1965 v l girko ital spectral theory of random matrices in russian nauka moscow 1988 and mahaux weidenm umlt u ller chaotic resonance scattering c mahaux and h a weidenm umlt u ller ital shell model approach to nuclear reactions north holland amsterdam 1969 copyright ital 1997 ital the american physical society

generalized swanson models and their solutions we analyse a class of non hermitian quadratic hamiltonians which are of the form where are real constants with and and are generalized creation and annihilation operators respectively thus these hamiltonians may be classified as generalized swanson models it is shown that the eigenenergies are real for a certain range of values of the parameters a similarity transformation mapping the non hermitian hamiltonian h to a hermitian one h is also obtained it is shown that h and h share identical energies as explicit examples the solutions of a couple of models based on the trigonometric rosen morse i and the hyperbolic rosen morse ii type potentials are obtained we also study the case where the non hermitian hamiltonian is symmetric

large scale correlations in normal non hermitian matrix ensembles we compute the large scale macroscopic correlations in ensembles of normal random matrices with a general non gaussian measure and in ensembles of general non hermitian matrices with a class of non gaussian measures in both cases the eigenvalues are complex and in the large n limit they occupy a domain in the complex plane for the case when the support of eigenvalues is a connected compact domain we compute two three and four point connected correlation functions in the first non vanishing order in 1 n in a manner that the algorithm of computing higher correlations becomes clear the correlation functions are expressed through the solution of the dirichlet boundary problem in the domain complementary to the support of eigenvalues

logarithmic moments of characteristic polynomials of random matrices abstract in a recent article we have discussed the connections between averages of powers of riemann s function on the critical line and averages of characteristic polynomials of random matrices the result for random matrices was shown to be universal i e independent of the specific probability distribution and the results were derived for arbitrary moments this allows one to extend the previous results to logarithmic moments for which we derive the explicit universal expressions in random matrix theory we then compare these results to various results and conjectures for functions and the correspondence is again striking

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

non hermitian euclidean random matrix theory we develop a theory for the eigenvalue density of arbitrary non hermitian euclidean matrices closed equations for the resolvent and the eigenvector correlator are derived the theory is applied to the random green s matrix relevant to wave propagation in an ensemble of pointlike scattering centers this opens a new perspective in the study of wave diffusion anderson localization and random lasing

statistics of the eigenvalues of tsallis matrices abstract the statistics of the eigenvalues of symmetric random matrices composed by real and statistically independent elements following the distribution that maximizes tsallis s entropy is carried numerically in the limit of large matrices for entropic indexes in the interval 5 3 by using a convenient rescale of variables it is possible to show that such matrices fall in the same class of the gaussian orthogonal ensemble goe for the entropic index 5 3 q the density of eigenvalues and the distribution of level spacings do not seem to follow a simple rescale of variables involving different values of q and exhibit a behavior very distinct from the goe both quantities present long tails for 5 3 q 2 and such long tails die out when q 2 the density of eigenvalues appears to be symmetric around zero exhibiting a peak at the origin that becomes steeper for increasing values of q approaching a delta function at the origin when q 3 the distribution of level spacings displays a form that resembles the well known wigner s surmise augmented by a long tail for q slightly greater than 5 3 but gets deformed for increasing values of q approaching an exponential decay for q 2 for q close to 3 our results resemble those of very sparse random matrices characterized by many zero matrix elements

breakdown of universality in multi cut matrix models we solve the puzzle of the disagreement between orthogonal polynomials methods and mean field calculations for random n n matrices with a disconnected eigenvalue support we show that the difference does not stem from a 2 symmetry breaking but from the discreteness of the number of eigenvalues this leads to additional terms quasiperiodic in n which must be added to the naive mean field expressions our result invalidates the existence of a smooth topological large n expansion and some postulated universality properties of correlators we derive the large n expansion of the free energy for the general two cut case from it we rederive by a direct and easy mean field like method the two point correlators and the asymptotic orthogonal polynomials we extend our results to any number of cuts and to non real potentials

truncated linear statistics associated with the eigenvalues of random matrices ii partial sums over proper time delays for chaotic quantum dots invariant ensembles of random matrices are characterized by the distribution of their eigenvalues lambda 1 ldots lambda n we study the distribution of truncated linear statistics of the form tilde l sum i 1 p f lambda i with p n this problem has been considered by us in a previous paper when the p eigenvalues are further constrained to be the largest ones or the smallest in this second paper we consider the same problem without this restriction which leads to a rather different analysis we introduce a new ensemble which is related but not equivalent to the thinned ensembles introduced by bohigas and pato this question is motivated by the study of partial sums of proper time delays in chaotic quantum dots which are characteristic times of the scattering process using the coulomb gas technique we derive the large deviation function for tilde l large deviations of linear statistics l sum i 1 n f lambda i are usually dominated by the energy of the coulomb gas which scales as sim n 2 implying that the relative fluctuations are of order 1 n for the truncated linear statistics considered here there is a whole region including the typical fluctuations region where the energy of the coulomb gas is frozen and the large deviation function is purely controlled by an entropic effect because the entropy scales as sim n the relative fluctuations are of order 1 sqrt n our analysis relies on the mapping on a problem of p fictitious non interacting fermions in n energy levels which can exhibit both positive and negative effective absolute temperatures we determine the large deviation function characterizing the distribution of the truncated linear statistics and show that for the case considered here f lambda 1 lambda the corresponding phase diagram is separated in three different phases

eigenvalue distributions of large euclidean random matrices for waves in random media we study probability distributions of eigenvalues of hermitian and non hermitian euclidean random matrices that are typically encountered in the problems of wave propagation in random media

rotational invariant estimator for general noisy matrices we investigate the problem of estimating a given real symmetric signal matrix c from a noisy observation matrix m in the limit of large dimension we consider the case where the noisy measurement m comes either from an arbitrary additive or multiplicative rotational invariant perturbation we establish using the replica method the asymptotic global law estimate for three general classes of noisy matrices significantly extending previously obtained results we give exact results concerning the asymptotic deviations called overlaps of the perturbed eigenvectors away from the true ones and we explain how to use these overlaps to clean the noisy eigenvalues of m we provide some numerical checks for the different estimators proposed in this paper and we also make the connection with some well known results of bayesian statistics

s matrix theory for transmission through billiards in tight binding approach in the tight binding approximation we consider multi channel transmission through a billiard coupled to leads following dittos we derive the coupling matrix the scattering matrix and the effectiv

stable transition layers in a balanced bistable equation with degeneracy abstract in this paper we consider the semilinear elliptic problem e2uxx u x 2 u2 in 0 1 with the neumann boundary condition the function is positive and is allowed to degenerate at where takes its local minimum we construct a stable solution which has exactly one layer near the interval where degenerates