anderson transition in a three dimensional kicked rotor we investigate anderson localization in a three dimensional 3d kicked rotor by a finite size scaling analysis we identify a mobility edge for a certain value of the kicking strength k k c for k k c dynamical localization does not occur all eigenstates are delocalized and the spectral correlations are well described by wigner dyson statistics this can be understood by mapping the kicked rotor problem onto a 3d anderson model am where a band of metallic states exists for sufficiently weak disorder around the critical region k approximately k c we carry out a detailed study of the level statistics and quantum diffusion in agreement with the predictions of the one parameter scaling theory opt and with previous numerical simulations the number variance is linear level repulsion is still observed and quantum diffusion is anomalous with proportional t 2 3 we note that in the 3d kicked rotor the dynamics is not random but deterministic in order to estimate the differences between these two situations we have studied a 3d kicked rotor in which the kinetic term of the associated evolution matrix is random a detailed numerical comparison shows that the differences between the two cases are relatively small however in the deterministic case only a small set of irrational periods was used a qualitative analysis of a much larger set suggests that deviations between the random and the deterministic kicked rotor can be important for certain choices of periods heuristically it is expected that localization effects will be weaker in a nonrandom potential since destructive interference will be less effective to arrest quantum diffusion however we have found that certain choices of irrational periods enhance anderson localization effects

extended convergence regions for the aor method abstract in this paper we give sufficient conditions for the convergence of the aor method when the matrix a for ax b is a strictly diagonally dominant matrix these results improve the conclusions obtained in the theorem 4 10 with the notion of generalized diagonal dominant matrix we enlarge the convergence regions given in theorem 9 10 when a is a nonsingular h matrix in the last section we generalize theorem 6 of robert 11 and we present some results which extend the convergence regions for the aor method

comparison results for parallel multisplitting methods with applications to aor methods abstract in this paper we present some comparison theorems between two general parallel multisplittings these comparison theorems can be applied to many classical splittings such as jacobi gauss seidel and aor splittings some significant improvements and generalizations of the existing comparison results are obtained

convergence theory for the general gaor type iterative method and the msor iterative method applied to h matrices abstract we discuss the general gaor type ggaor iterative method of which special cases are the aor the gaor and the msor iterative methods to solve the linear system ax b where a i l u with l and u being general matrices the ggaor iteration matrix is expressed by l r i rl 1 i r l u where r and are diagonal matrices and the msor iteration matrix l i 1 i u some basic results on the upper bounds of the spectral radii l r and l are given when a is strictly or irreducibly diagonally dominant by rows based upon these results we obtain new results on the convergence regions of the ggaor and the msor iterative methods when a is an h matrix or a has property a and recover and improve previous ones

critical statistics for non hermitian matrices we introduce a generalized ensemble of non hermitian matrices interpolating between the gaussian unitary ensemble the ginibre ensemble and the poisson ensemble the joint eigenvalue distribution of this model is obtained by means of an extension of the itzykson zuber formula to general complex matrices its correlation functions are studied both in the case of weak non hermiticity and in the case of strong non hermiticity in the weak non hermiticity limit we show that the spectral correlations in the bulk of the spectrum display critical statistics the asymptotic linear behavior of the number variance is already approached for energy differences of the order of the eigenvalue spacing to lowest order its slope does not depend on the degree of non hermiticity close the edge the spectral correlations are similar to the hermitian case in the strong non hermiticity limit the crossover behavior from the ginibre ensemble to the poisson ensemble first appears close to the surface of the spectrum our model may be relevant for the description of the spectral correlations of an open disordered system close to an anderson transition

strong convergence to global solutions for multidimensional flows of compressible viscous fluids with polytropic equations of state and discontinuous initial data we extend to general polytropic pressures p k 1 the existence theory of 8 for isothermal 1 flows of navier stokes fluids in two and three space dimensions with fairly general initial data specifically we require that the initial density be close to a constant in l2 and l and that the initial velocity be small in l2 and bounded in l2n in two dimensions the l2 norms must be weighted slightly solutions are obtained as limits of approximate solutions corresponding to mollified initial data the key point is that the approximate densities are shown to converge strongly so that nonlinear pressures can be accommodated even in the absence of any uniform regularity information for the approximate densities

decay estimates of solutions for the equations of motion of compressible viscous and heat conductive gases in an exterior domain in ℝ3 we consider the equations of motion of compressible viscous and heat conductive gases in an exterior domain in 3 we give the l q l p estimates for solutions to the linearized equations and show an optimal decay estimate for solutions to the nonlinear problem

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

convergence for a general form of the gaor method and its application to the msor method abstract two types of gaor iterative methods were respectively given by a hadjidimos and hu jia gan to solve the linear system ax b where a d e f with d to be nonsingular and diagonal in the former method the parameters involved are two numbers and and in the latter method two diagonal matrices r and the msor iterative method given by d m young is a special case of the latter gaor method for uniformity of the results the paper considers the convergence for a general form of the aor type iteration matrix l r i rl 1 i r l u where l d 1e and u d 1f and gives some important results for the convergence domain of lr when a d e f is an h matrix applying the results to the gaor aor gsor and msor methods we recover certain recent main results and greatly extend certain other previous results

a survey of the iterative methods for the solution of linear systems by extrapolation relaxation and other techniques abstract for the iterative solution of linear systems of algebraic equations ax b 1 with a c n n x b c n and det a 0 numerous methods exist although a classification of them seems not to be possible one may note that the first step for the construction of an iterative method usually begins with a splitting of a in 1 thus a is written as a m n where det m 0 and m is easily inverted so that 1 is equivalent to x tx c 2 t m 1n c m 1 the discussion will be restricted to the so called linear stationary iterative methods although some nonstationary ones will be mentioned and emphasis will be given to those which fall into the categories of extrapolation relaxation and similar ones for this it will be assumed that the spectrum t of t in 2 is contained in a well defined compact region r whose complement with respect to the complex plane is simply connected and 1 r under these assumptions and for each specific class of methods described each time an attempt will be made to present the optimum one the optimum is the one out of the class of methods for which the sequence of vectors yielded converges asymptotically to the unique solution of 1 as fast as possible

diluted banded random matrices scaling behavior of eigenfunction and spectral properties we demonstrate that the normalized localization length of the eigenfunctions of diluted sparse banded random matrices follows the scaling law the scaling parameter of the model is defined as where is the average number of non zero elements per matrix row n is the matrix size and additionally we show that also scales the spectral properties of the model up to certain sparsity characterized by the spacing distribution of eigenvalues

multifractality of one electron eigen states in 1d disordered long range models abstract one investigates the spatial structure of one electron eigenstates for the one dimensional anderson model with long range off diagonal disorder h n m 1 1 n m where a b means a uniform random distribution between a and b two cases are considered according to the choices for the on site energy e n namely e n 0 and e n 0 1 for 1 all states are critical and the multifractal spectra f is computed for the states near the center of the band the level spacing distribution function is also obtained and its behavior for both small and large separations are studied

family of solvable generalized random matrix ensembles with unitary symmetry we construct a very general family of characteristic functions describing random matrix ensembles rme having a global unitary invariance and containing an arbitrary one variable probability measure which we characterize by a spread function various choices of the spread function lead to a variety of possible generalized rmes which show deviations from the well known gaussian rme originally proposed by wigner we obtain the correlation functions of such generalized ensembles exactly and show examples of how particular choices of the spread function can describe ensembles with arbitrary eigenvalue densities as well as critical ensembles with multifractality

eigenvalue correlations for generalized gaussian ensembles abstract we study the statistical properties of the eigenvalues of a generalized gaussian ensemble of hermitian matrices described by a set of the variances of the matrix elements and non invariant under an unitary transformation we find that the eigenvalue distribution in this case can be mapped on to the corresponding distribution in dyson s brownian ensembles with a function of variances in the former playing the role of time in the latter the pre existing information about the spectral correlations of brownian ensembles can therefore be used to obtain the same for generalized gaussian ensembles

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

note on irreducible diagonally dominant matrices and the convergence of the aor iterative method considering the linear systems ax b where the matrix a is irreducible and diagonally dominant we obtain bounds for the spectral radius of the 4 matrix of the aor method and we achieve the convergence conditions given in 2 by a different method if a is strictly diagonally dominant we get larger intervals for the parameter x of the sor method and we improve the results of theorems 5 6 of 3 for the aor method

energy level statistics of a critical random matrix ensemble we study energy level statistics of a critical random matrix ensemble of power law banded complex hermitian matrices we compute the level compressibility via the level number variance and compare it with the analytical formula for the exactly solvable model of moshe neuberger and shapiro

spectral statistics of random toeplitz matrices the spectral statistics of hermitian random toeplitz matrices with independent and identically distributed elements are investigated numerically it is found that eigenvalue statistics of complex toeplitz matrices are surprisingly well approximated by the semi poisson distribution belonging to intermediate type statistics observed in certain pseudointegrable billiards the origin of intermediate behavior could be attributed to the fact that fourier transformed random toeplitz matrices have the same slow decay outside the main diagonal as critical random matrix ensembles the statistical properties of the full spectrum of real random toeplitz matrices are close to the poisson distribution but each of their constituent subspectra is again well described by the semi poisson distribution the findings indicate that intermediate statistics in general and the semi poisson distribution in particular are more universal than considered before

eigenfunction statistics of complex systems a common mathematical formulation we derive a common mathematical formulation for the eigenfunction statistics of hermitian operators represented by a multiparametric probability density the system information in the formulation enters through two parameters only namely system size and the complexity parameter a function of all system parameters including size the behavior is contrary to the eigenvalue statistics which is sensitive to the complexity parameter only and shows a single parametric scaling the existence of a mathematical formulation of both eigenfunctions and eigenvalues common to a wide range of complex systems indicates the possibility of a similar formulation for many physical properties this also suggests the possibility to classify them in various universality classes defined by the complexity parameter

an iterative method with truncation for infinite linear systems abstract for the infinite linear system tx v where t is an infinite uniformly strictly diagonally dominant matrix and x and v are infinite vectors sufficient conditions are imposed on t and v for the existence and uniqueness of a bounded solution the system is rewritten as x ax b which is first iterated and then truncated to a finite system convergence of the solution of the resulting system to that of the original system is established and explicit error bounds are determined examples are given to illustrate the computation

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

asymptotic behavior of solutions of the compressible navier stokes equations on the half space asymptotic behavior of solutions of the compressible navier stokes equations on the half space r n n 2 is considered around a given constant equilibrium a solution formula for the linearized problem is derived and l p estimates for solutions of the linearized problem are obtained for 2 p it is shown that as in the case of the cauchy problem the leading part of the solution of the linearized problem is decomposed into two parts one that behaves like diffusion waves and the other one purely diffusively there however are some aspects different from the cauchy problem especially in considering spatial derivatives it is also shown that the solution of the linearized problem approaches for large times the solution of the nonstationary stokes problem in some l p spaces and as a result a solution formula for the nonstationary stokes problem is obtained large time behavior of solutions of the nonlinear problem is then investigated in l p spaces for 2 p by applying the results on the linearized analysis and the weighted energy method the results indicate that there may be some nonlinear interaction phenomena not appearing in the cauchy problem

on parameterized inexact uzawa methods for generalized saddle point problems abstract for the large sparse saddle point problems bai et al recently studied a class of parameterized inexact uzawa piu methods z z bai b n parlett z q wang on generalized successive overrelaxation methods for augmented linear systems numer math 102 2005 1 38 for a special case that the 1 1 block is solved exactly they determined the convergence domain and computed the optimal iteration parameters and the corresponding optimal convergence factor for the induced method in this paper we develop these methods to the large sparse generalized saddle point problems for the obtained parameterized inexact uzawa method we prove its convergence under suitable restrictions on the iteration parameters in particular we determine its quasi optimal iteration parameters and the corresponding quasi optimal convergence factor for the saddle point problems furthermore this piu method is generalized to obtain a framework of accelerated variants of the parameterized inexact uzawa methods for solving both symmetric and nonsymmetric generalized saddle point problems by using the techniques of vector extrapolation matrix relaxation and inexact iteration

return probability and scaling exponents in the critical random matrix ensemble we study an asymptotic behavior of the return probability for the critical random matrix ensemble in the regime of strong multifractality the return probability is expected to show critical scaling in the limit of large time or large system size using the supersymmetric virial expansion we confirm the scaling law and find analytical expressions for the fractal dimension of the wavefunctions d2 and the dynamical scaling exponent by comparing them we verify the validity of chalker s ansatz for dynamical scaling

spectral fluctuations of billiards with mixed dynamics from time series to superstatistics a statistical analysis of the eigenfrequencies of two sets of superconducting microwave billiards one with mushroomlike shape and the other from the family of the limacon billiards is presented these billiards have mixed regular chaotic dynamics but different structures in their classical phase spaces the spectrum of each billiard is represented as a time series where the level order plays the role of time two most important findings follow from the time series analysis first the spectra can be characterized by two distinct relaxation lengths this is a prerequisite for the validity of the superstatistical approach which is based on the folding of two distribution functions second the shape of the resulting probability density function of the so called superstatistical parameter is reasonably approximated by an inverse chi2 distribution this distribution is used to compute nearest neighbor spacing distributions and compare them with those of the resonance frequencies of billiards with mixed dynamics within the framework of superstatistics the obtained spacing distribution is found to present a good description of the experimental ones and is of the same or even better quality as a number of other spacing distributions including the one from berry and robnik however in contrast to other approaches toward a theoretical description of spectral properties of systems with mixed dynamics superstatistics also provides a description of properties of the eigenfunctions in terms of a superstatistical generalization of the porter thomas distribution indeed the inverse chi2 parameter distribution is found suitable for the analysis of experimental resonance strengths in the limacon billiards within the framework of superstatistics

a survey of aor and tor methods abstract this paper makes a survey on the aor and tor methods we present the recent results about convergence and determination of the optimum parameters mainly those obtained by chinese authors

power law eigenvalue density scaling and critical random matrix ensembles we consider a class of rotationally invariant unitary random matrix ensembles where the eigenvalue density falls off as an inverse power law under a scaling appropriate for such power law densities different from the scaling required in gaussian random matrix ensembles we calculate exactly the two level kernel that determines all eigenvalue correlations we show that such ensembles belong to the class of critical ensembles

global solutions of the navier stokes equations for multidimensional compressible flow with discontinuous initial data abstract we prove the global existence of weak solutions of the navier stokes equations for compressible isothermal flow in two and three space dimensions when the initial density is close to a constant in l 2 and l and the initial velocity is small in l 2 and bounded in l 2 n in two dimensions the l 2 norms must be weighted slightly a great deal of qualitative information about the solution is obtained for example we show that the velocity and vorticity are relatively smooth in positive time as is the effective viscous flux f which is the divergence of the velocity minus a certain multiple of the pressure we find that f plays a crucial role in the entire analysis particularly in closing the required energy estimates understanding rates of regularization near the initial layer and most important obtaining time independent pointwise bounds for the density

on generalized successive overrelaxation methods for augmented linear systems for the augmented system of linear equations golub wu and yuan recently studied an sor like method bit 41 2001 71 85 by further accelerating it with another parameter in this paper we present a generalized sor gsor method for the augmented linear system we prove its convergence under suitable restrictions on the iteration parameters and determine its optimal iteration parameters and the corresponding optimal convergence factor theoretical analyses show that the gsor method has faster asymptotic convergence rate than the sor like method also numerical results show that the gsor method is more effective than the sor like method when they are applied to solve the augmented linear system this gsor method is further generalized to obtain a framework of the relaxed splitting iterative methods for solving both symmetric and nonsymmetric augmented linear systems by using the techniques of vector extrapolation matrix relaxation and inexact iteration besides we also demonstrate a complete version about the convergence theory of the sor like method

generalized stein rosenberg theorems for the regular splittings and convergence of some generalized iterative methods we study the convergence properties of the aor and gaor iterative methods for the solution of the linear system ax b and we give more general forms of the stein rosenberg theorem under the assumptions that 1 l u 0 in a i b i l u are not necessarily strictly lower and strictly upper triangular matrices respectively and 2 l 1 we derive new results regarding the spectral radii of the gaor or aor and the gj or jacobi iteration matrices and give in the plane sufficient conditions domains for the gaor and aor method to converge

power law random banded matrices and ultrametric matrices eigenvector distribution in the intermediate regime the power law random banded matrices and the ultrametric random matrices are investigated numerically in the regime where eigenstates are extended but all integer matrix moments remain finite in the limit of large matrix dimensions though in this case standard analytical tools are inapplicable we found that in all considered cases eigenvector distributions are extremely well described by the generalised hyperbolic distribution which differs considerably from the usual porter thomas distribution but shares with it certain universal properties

eigenfunction distribution for the rosenzweig porter model the statistical distribution of eigenfunctions for the rosenzweig porter model is derived for the region where eigenfunctions have fractal behaviour the result is based on simple physical ideas and leads to transparent explicit formulas which agree very well with numerical calculations it constitutes a rare case where a non trivial eigenfunction distribution is obtained in a closed form

spacing distributions for real symmetric 2 2 generalized gaussian ensembles for ensembles of 2 2 real symmetric matrices the normalized spacing distributions p s form a family whose parameter space is the unit cube with coordinates determined by the means and variances of the diagonal and off diagonal elements the cube contains a variety of spacing distributions that are calculated analytically they include the wigner poisson transition distributions with singularities and gaussians unfolding is superfluous for 2 2 matrices but it can be implemented giving rise to a further variety of spacing distributions some surprising

critical statistics in quantum chaos and calogero sutherland model at finite temperature we investigate the spectral properties of a generalized gaussian orthogonal ensemble capable of describing critical statistics the joint distribution of eigenvalues of this model is expressed as the diagonal element of the density matrix of a gas of particles governed by the calogero sutherland cs hamiltonian taking advantage of the correspondence between cs particles and eigenvalues and utilizing a recently conjectured expression by kravtsov and tsvelik for the finite temperature density density correlations of the cs model we show that the number variance of our random matrix model is asymptotically linear with a slope depending on the parameters of the model such linear behavior is a signature of critical statistics this random matrix model may be relevant for the description of spectral correlations of complex quantum systems with a self similar or fractal poincare section of its classical counterpart this is shown in detail for two examples the anisotropic kepler problem and a kicked particle in a well potential in both cases the number variance and the delta 3 statistic are accurately described by our analytical results

global existence of small solutions to a class of nonlinear evolution equations on etudie le comportement global des solutions d une classe d equations d evolution non lineaires on considere le probleme aux valeurs initiales pour des equations quasilineaires qui sont des perturbations d equations lineaires dissipatives

generating a fractal butterfly floquet spectrum in a class of driven su 2 systems eigenstate statistics the floquet spectra of a class of driven su 2 systems have been shown to display butterfly patterns with multifractal properties the implication of such critical spectral behavior for the floquet eigenstate statistics is studied in this work following the methodologies for understanding the fractal behavior of energy eigenstates of time independent systems on the anderson transition point we analyze the distribution profile the mean value and the variance of the logarithm of the inverse participation ratio of the floquet eigenstates associated with multifractal floquet spectra the results show that the floquet eigenstates also display fractal behavior but with features markedly different from those in time independent anderson transition models this motivated us to propose random unitary matrix ensemble called power law random banded unitary matrix ensemble to illuminate the floquet eigenstate statistics of critical driven systems the results based on the proposed random matrix model are consistent with those obtained from our dynamical examples with or without time reversal symmetry

on the convergence of some generalized iterative methods abstract this paper is concerned with the generalized accelerated overrelaxation gaor method which constitutes a generalization of the basic iterative methods for the solution of linear systems a number of new theoretical results are presented concerning the convergence theory of the gaor method and special cases of it much attention is given to linear systems with positive definite coefficient matrices although the problem of determining the various optimum parameters of the gaor method in the general case is very difficult to solve theoretically and is still an open one the numerical examples which are presented in this paper show that a suitable exploitation of it may give much better results than the other basic iterative methods

initial boundary value problems for the equations of motion of compressible viscous and heat conductive fluids the equations of motion of compressible viscous and heat conductive fluids are investigated for initial boundary value problems on the half space and on the exterior domain of any bounded region the global solution in time is proved to exist uniquely and approach the stationary state ast provided the prescribed initial data and the external force are sufficiently small

generalized extrapolation principle and convergence of some generalized iterative methods abstract to solve the linear system ax b this paper presents a generalized extrapolated method by replacing the extrapolation parameter with the diagonal matrix and systematically gives the basic results for its convergence based upon these results the paper considers the convergence of the gj and gaor iterative methods and using the set of the equimodularized diagonally similar matrices defined here gives some new further convergence results for h matrices and their subclasses strictly or irreducibly diagonally dominant matrices which unify improve and extend previously given various results finally conditions equivalent to the statement that a is a nonsingular h matrix or a strictly or an irreducibly diagonally dominant matrix are given in connection with the gj and gaor methods

a supersymmetry approach to almost diagonal random matrices we develop a supersymmetric field theoretical description of the gaussian ensemble of the almost diagonal hermitian random matrices the matrices have independent random entries hi j with parametrically small off diagonal elements we derive a regular virial expansion of correlation functions in the number of interacting supermatrices associated with different sites in the real space and demonstrate that the perturbation theory constructed in this way is controlled by a small parameter the general form of the integral expression for the mth virial coefficient governed by the interaction of m supermatrices is presented and calculated explicitly in the cases of 2 and 3 matrix interaction the suggested technique allows us to calculate both the spectral correlations and the correlations of the eigenfunctions taken at different energies and in different space points

optimal decay rate for strong solutions in critical spaces to the compressible navier stokes equations abstract in this paper we are concerned with the convergence rates of the global strong solution to motionless state with constant density for the compressible navier stokes equations in the whole space r n for n 3 it is proved that the perturbations decay in critical spaces if the initial perturbations of density and velocity are small in b 2 1 n 2 r n b 1 0 r n and b 2 1 n 2 1 r n b 1 0 r n respectively

the nature of electronic states in a disordered chain with long ranged hopping amplitudes abstract in this work we investigate the nature of the electronic states in a one dimensional tight binding model with on site energetic disorder and long ranged hopping amplitudes these are assumed to follow a power law decay behavior with distance in the form h r 1 r with 1 direct numerical diagonalization finite size scaling and a spectral statistics analysis show that as 1 the localization length exhibits an exponential growth as the upper band edge is approached strong finite size corrections are present in this regime near the upper band edge with the one electron states displaying a strong level repulsion even at large although finite chains

on stein rosenberg type theorems for nonnegative and perron frobenius splittings abstract the stein rosenberg theorem is extended and generalized to the class of nonnegative splittings a m 1 n 1 m 2 n 2 as well as to the most generalized class of perron frobenius splittings two types of the stein rosenberg theorem are stated and proved for both classes these theorems allow us to obtain comparison results for the rate of convergence of the associated iterative methods specific assumptions are given under which the inequalities of the spectral radii become equalities or strict inequalities the theoretical results are confirmed by numerical examples

level compressibility in a critical random matrix ensemble the second virial coefficient we study spectral statistics of a gaussian unitary critical ensemble of almost diagonal hermitian random matrices with off diagonal entries hij 2 b2 i j 2 small compared to diagonal ones hii 2 1 using the recently suggested method of virial expansion in the number of interacting energy levels yevtushenko and kravtsov 2003 j phys a math gen 36 8265 we calculate a coefficient b2 1 in the level compressibility b we demonstrate that only the leading terms in b coincide for this model and for an exactly solvable model suggested by moshe et al 1994 phys rev lett 73 1497 the sub leading terms b2 being different numerical data confirm our analytical calculation

on asymptotic expansions and scales of spectral universality in band random matrix ensembles we consider real random symmetric n n matrices h of the band type form with characteristic length b the matrix entries are independent gaussian random variables and have the variance proportional to where u t vanishes at infinity we study the resolvent in the limit and obtain the explicit expression for the leading term of the first correlation function of the normalized trace

rotationally invariant family of levy like random matrix ensembles we introduce a family of rotationally invariant random matrix ensembles characterized by a parameter while 1 corresponds to well known critical ensembles we show that 1 describes levy like ensembles characterized by power law eigenvalue densities for 1 the density is bounded as in gaussian ensembles but 1 describes ensembles characterized by densities with long tails in particular the model allows us to evaluate in terms of a novel family of orthogonal polynomials the eigenvalue correlations for levy like ensembles these correlations differ qualitatively from those in either the gaussian or the critical ensembles