universal correlations and power law tails in financial covariance matrices abstract we investigate whether quantities such as the global spectral density or individual eigenvalues of financial covariance matrices can be best modelled by standard random matrix theory or rather by its generalisations displaying power law tails in order to generate individual eigenvalue distributions a chopping procedure is devised which produces a statistical ensemble of asset price covariances from a single instance of financial data sets local results for the smallest eigenvalue and individual spacings are very stable upon reshuffling the time windows and assets they are in good agreement with the universal tracy widom distribution and wigner surmise respectively this suggests a strong degree of robustness especially in the low lying sector of the spectra most relevant for portfolio selections conversely the global spectral density of a single covariance matrix as well as the average over all unfolded nearest neighbour spacing distributions deviate from standard gaussian random matrix predictions the data are in fair agreement with a recently introduced generalised random matrix model with correlations showing a power law decay

topological abstraction of higher dimensional automata higher dimensional automata constitute a very expressive model for concurrent systems in this paper we discuss topological abstraction of higher dimensional automata i e the replacement of hdas by smaller ones that can be considered equivalent from the point of view of both computer science and topology by definition topological abstraction preserves the homotopy type the trace category and the homology graph of an hda we establish conditions under which cube collapses yield topological abstractions of hdas

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

noise dressing of financial correlation matrices we show that results from the theory of random matrices are potentially of great interest to understand the statistical structure of the empirical correlation matrices appearing in the study of multivariate time series the central result of the present study which focuses on the case of financial price fluctuations is the remarkable agreement between the theoretical prediction based on the assumption that the correlation matrix is random and empirical data concerning the density of eigenvalues associated to the time series of the different stocks of the s 500 or other major markets in particular the present study raises serious doubts on the blind use of empirical correlation matrices for risk management

finite n fluctuation formulas for random matrices for the gaussian and laguerre random matrix ensembles the probability density function p d f for the linear statistic jn 1 xj x is computed exactly and shown to satisfy a central limit theorem asn for the circular random matrix ensemble the p d f s for the statistics jn 1 j and jn 1 log 2 sin j 2 are calculated exactly by using a constant term identity from the theory of the selberg integral and are also shown to satisfy a central limit theorem asn

some consequences of exchangeability in random matrix theory properties of infinite sequences of exchangeable random variables result directly in explicit expressions for calculating asymptotic densities of eigenvalues of any ensemble of random matrices h whose distribution depends only on tr h h where h is the hermitian conjugate of h for real symmetric matrices and for hermitian matrices the densities are constructed by summing up wigner semicircles with varying radii and weights as confirmed by monte carlo simulations extensions to more general matrix ensembles are also considered

the non sequential behaviour of petri nets the idea of representing non sequential processes as partially ordered sets occurrence nets is applied to place transition nets petri nets based on the well known notion of process for condition event systems for occurrence nets some theorems relating k density cut finiteness and discreteness are proved with these theorems the result that a place transition net is bounded if and only if its processes are k dense is obtained

semiclassical level spacings when regular and chaotic orbits coexist the authors calculate semiclassical limiting level spacing distributions p s for systems whose classical energy surface is divided into a number of separate region in which motion is regular or chaotic in the calculation it is assumed that the spectrum is the superposition of statistically independent sequences of levels from each of the classical phase space regions sequences from regular regions having poisson distributions and those from irregular regions having wigner distributions the formulae for p s depend on the sum of the liouville measures of all the classical regular regions and on the separate liouville measures of the significant chaotic regions

theory of levy matrices we investigate the statistical properties of the spectrum of large symmetrical matrices with each element mathit h mathit i mathit j chosen according to a broad distribution ensuremath rho h decaying for large h as mathit h mathrm ensuremath 1 mathrm ensuremath mathrm ensuremath mu for ensuremath mu g2 mathit h 2 is finite and the well known gaussian orthogonal ensemble goe results are recovered when ensuremath mu 2 the semicircular law is replaced by a density which extends over the whole energy axis furthermore while all states are extended in the case of goe matrices we show numerically and analytically that two mobility edges appear separating extended from localized states with an intermediate mixed phase in between the unusual nature of these localized states is discussed

the distributions of the determinant of fixed trace ensembles of real symmetric and of hermitian random matrices probability densities of the determinant are obtained for fixed trace ensembles of n x n hermitian and real symmetric matrices from the exact determinant densities of the gaussian orthogonal ensemble goe and of the gaussian unitary ensemble gue respectively

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

the fixed trace β hermite ensemble of random matrices and the low temperature distribution of the determinant of an n n β hermite matrix the hermite ensemble he of tridiagonal n n random matrices of dumitriu and edelman 2002 j math phys 43 5830 is a continuum of ensembles in which the reciprocal of the temperature in the 2d electrostatic interpretation of the eigenvalue characteristics can take any value the eigenvalue distributions coincide with those of the classical gaussian ensembles goe gue gse for 1 2 4 a fixed trace hermite ensemble fthe is defined from the he and is used to extend the spherical ensembles of classical symmetries to spherical ensembles at low temperature when for a fixed value of n the asymptotic distributions of reduced determinants dn of random n n h and fth matrices are shown to be standard gaussians accordingly the fluctuations of the potential at the origin ln dn have a generalized gumbel distribution at low temperature for large n and large a ln n variance results from the strongly correlated fluctuations of eigenvalues around their equilibrium positions

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

random matrix ensembles from nonextensive entropy the classical gaussian ensembles of random matrices can be constructed by maximizing boltzmann gibbs shannon s entropy s text bgs ensuremath ensuremath int d mathbf h p mathbf h mathrm ln p mathbf h with suitable constraints here we construct and analyze random matrix ensembles arising from the generalized entropy s q 1 ensuremath ensuremath int d mathbf h p mathbf h q q ensuremath 1 thus s 1 s text bgs the resulting ensembles are characterized by a parameter q measuring the degree of nonextensivity of the entropic form making q ensuremath rightarrow 1 recovers the gaussian ensembles if q ensuremath ne 1 the joint probability distributions p mathbf h cannot be factorized i e the matrix elements of mathbf h are correlated in the limit of large matrices two different regimes are observed when ql1 p mathbf h has compact support and the fluctuations tend asymptotically to those of the gaussian ensembles anomalies appear for qg1 both p mathbf h and the marginal distributions p h ij show power law tails numerical analyses reveal that the nearest neighbor spacing distribution is also long tailed not wigner dyson and after proper scaling very close to the result for the 2 ifmmode times else texttimes fi 2 case char22 a generalization of wigner s surmise we discuss connections of these nonextensive ensembles with other non gaussian ones such as the so called l evy ensembles and those arising from soft confinement

normalization sum rule and spontaneous breaking of u n invariance in random matrix ensembles it is shown that the two level correlation function mathit r 2 mathrm ensuremath infty s mathit s mathcal in the invariant random matrix ensembles rme s with soft confinement exhibits unexpected long range correlations in the form of a sharp peak at s ensuremath approxeq s the presence of such a ghost correlation peak lifts the sum rule prohibition for the level number variance to have a poisson like term var n ensuremath eta n that is typical of rme s with broken u n symmetry thus we conclude that the u n invariance is broken spontaneously in the rme s with soft confinement ensuremath eta playing the role of an order parameter

family of generalized random matrix ensembles using the generalized maximum entropy principle based on the nonextensive q entropy a family of random matrix ensembles is generated this family unifies previous extensions of random matrix theory rmt and gives rise to an orthogonal invariant stable levy ensemble with new statistical properties some of them are analytically derived

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

free random levy matrices using the theory of free random variables and the coulomb gas analogy we construct stable random matrix ensembles that are random matrix generalizations of the classical one dimensional stable levy distributions we show that the resolvents for the corresponding matrices obey transcendental equations in the large size limit we solve these equations in a number of cases and show that the eigenvalue distributions exhibit levy tails for the analytically known levy measures we explicitly construct the density of states using the method of orthogonal polynomials we show that the levy tail distributions are characterized by a different novel form of microscopic universality

configuration structures event structures and petri nets in this paper the correspondence between safe petri nets and event structures due to nielsen plotkin and winskel is extended to arbitrary nets without self loops under the collective token interpretation to this end we propose a more general form of event structure matching the expressive power of such nets these new event structures and nets are connected by relating both notions with configuration structures which can be regarded as representations of either event structures or nets that capture their behaviour in terms of action occurrences and the causal relationships between them but abstract from any auxiliary structure a configuration structure can also be considered logically as a class of propositional models or equivalently as a propositional theory in disjunctive normal from converting this theory to conjunctive normal form is the key idea in the translation of such a structure into a net for a variety of classes of event structures we characterise the associated classes of configuration structures in terms of their closure properties as well as in terms of the axiomatisability of the associated propositional theories by formulae of simple prescribed forms and in terms of structural properties of the associated petri nets

a modular analogue of morozov s theorem on maximal subalgebras of simple lie algebras abstract let g be a simple algebraic group over an algebraically closed field of characteristic p 0 and suppose that p is a very good prime for g in this paper we prove that any maximal lie subalgebra m of g lie g with rad m 0 has the form m lie p for some maximal parabolic subgroup p of g this means that morozov s theorem on maximal subalgebras is valid under mild assumptions on g we show that such assumptions are necessary by providing a counterexample to morozov s theorem for groups of type e 8 over fields of characteristic 5 our proof relies on the main results and methods of the classification theory of finite dimensional simple lie algebras over fields of prime characteristic

on invariant 2 2 β ensembles of random matrices abstract we introduce and solve exactly a family of invariant 2 2 random matrices depending on one parameter and we show that rotational invariance and real dyson index are not incompatible properties the probability density for the entries contains a weight function and a multiple trace trace interaction term which corresponds to the representation of the vandermonde squared coupling on the basis of power sums as a result the effective dyson index eff of the ensemble can take any real value in an interval two weight functions gaussian and non gaussian are explored in detail and the connections with ensembles of dumitriu edelman and the so called poisson wigner crossover for the level spacing are respectively highlighted a curious spectral twinning between ensembles of different symmetry classes is unveiled as a consequence the identification between symmetry group orthogonal unitary or symplectic and the exponent of the vandermonde 1 2 4 is shown to be potentially deceptive the proposed technical tool more generically allows for designing actual matrix models which i are rotationally invariant ii have a real dyson index eff iii have a pre assigned confining potential or alternatively level spacing profile the analytical results have been checked through numerical simulations with an excellent agreement eventually we discuss possible generalizations and further directions of research

random matrix theory within superstatistics we propose a generalization of the random matrix theory following the basic prescription of the recently suggested concept of superstatistics spectral characteristics of systems with mixed regular chaotic dynamics are expressed as weighted averages of the corresponding quantities in the standard theory assuming that the mean level spacing itself is a stochastic variable we illustrate the method by calculating the level density the nearest neighbor spacing distributions and the two level correlation functions for systems in transition from order to chaos the calculated spacing distribution fits the resonance statistics of random binary networks obtained in a recent numerical experiment

global spectrum fluctuations for the β hermite and β laguerre ensembles via matrix models we study the global spectrum fluctuations for hermite and laguerre ensembles via the tridiagonal matrix models introduced previously by the present authors j math phys 43 5830 2002 and prove that the fluctuations describe a gaussian process on polynomials we extend our results to slightly larger classes of random matrices

universal and nonuniversal properties of cross correlations in financial time series we use methods of random matrix theory to analyze the cross correlation matrix c of stock price changes of the largest 1000 u s companies for the 2 year period 1994 1995 we find that the statistics of most of the eigenvalues in the spectrum of c agree with the predictions of random matrix theory but there are deviations for a few of the largest eigenvalues we find that c has the universal properties of the gaussian orthogonal ensemble of random matrices furthermore we analyze the eigenvectors of c through their inverse participation ratio and find eigenvectors with large ratios at both edges of the eigenvalue spectrum a situation reminiscent of localization theory results

new family of unitary random matrices we consider an exactly solvable random matrix model related to the random transfer matrix model for disordered conductors in the conventional random matrix models the spacing distribution of nearest neighbor eigenvalues when expressed in units of average spacing has a universal behavior known generally as the wigner distribution in contrast our model has a single parameter as a function of which the spacing distribution crosses over from a wigner to a distribution which is increasingly more poisson like a feature common to a wide variety of physical systems including disorder and chaos

energy level statistics of an integrable billiard system in a rectangle in the hyperbolic plane a separable billiard system in the hyperbolic strip a particular realization of the hyperbolic plane is investigated the author considers a rectangle bounded by straight lines in the hyperbolic strip as an approximation of either a regular octagon with area a 4 pi or of a hyperbolic rectangle bounded by geodesics respectively due to the simple geometry of the approximation the schrodinger equation is separable and leads to simple transcendental equations for the eigenvalues in terms of legendre functions the statistical properties of the eigenvalue spectrum are investigated and checked with respect to weyl s 1912 law level spacing number variance and spectral rigidity respectively in particular the numerical results for the spectral rigidity are in close agreement with the semi classical theory of berry 1985

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

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

deformations of the tracy widom distribution in random matrix theory the tracy widom tw distribution describes the behavior of the largest eigenvalue we consider here two models in which tw undergoes transformations in the first one disorder is introduced in the gaussian ensembles by superimposing an external source of randomness a competition between tw and a normal gaussian distribution results depending on the spreading of the disorder the second model consists of removing at random a fraction of correlated eigenvalues of a random matrix the usual formalism of fredholm determinants extends naturally a continuous transition from tw to the weilbull distribution characteristic of extreme values of an uncorrelated sequence is obtained

a matrix model for the β jacobi ensemble this note presents a random matrix model for general 0 jacobi ensembles this generalizes the well known manova models for 1 2 4 and eliminates the quantization of and other parameters present in the previously known models this model is a partial answer to an open problem presented by dumitriu and edelman where they also presented models for the laguerre and hermite ensembles

exact densities of states of fixed trace ensembles of random matrices the densities of states and the associated characteristic functions of fixed trace ensembles ftes of n n random matrices mn tr mn mn constant are calculated exactly at finite n in the case of real symmetric matrices and of hermitian matrices the exact radial density is calculated at finite n for fixed trace ensembles of complex matrices with no further restrictions on the entries the density calculated in the hermitian case coincides with very recent literature results the exact finite n density of states of any ensemble of n n random matrices of a given symmetry whose probability density depends only on tr sn sn is simply obtained from the density of the fte of the same symmetry by a one dimensional integral

scaling and eigenfunction correlations near a mobility edge abstract the scaling form of the frequency and wavevector dependent diffusion constant is discussed near a mobility edge the consequences are stressed of fractal eigenfunction fluctuations for behaviour in the low frequency large wavevector regime

matrix models for beta ensembles this paper constructs tridiagonal random matrix models for general 0 hermite gaussian and laguerre wishart ensembles these generalize the well known gaussian and wishart models for 1 2 4 furthermore in the cases of the laguerre ensembles we eliminate the exponent quantization present in the previously known models we further discuss applications for the new matrix models and present some open problems

calculation of some determinants using the s shifted factorial several determinants with gamma functions as elements are evaluated these kinds of determinants are encountered for example in the computation of the probability density of the determinant of random matrices the s shifted factorial is defined as a generalization for non negative integers of the power function the rising factorial or pochammer s symbol and the falling factorial it is a special case of a polynomial sequence of the binomial type studied in combinatorics theory in terms of the gamma function an extension is defined for negative integers and even complex values properties mainly composition laws and binomial formulae are given they are used to evaluate families of generalized vandermonde determinants with s shifted factorials as elements instead of power functions

classification of balanced toral elements of exceptional lie algebras abstract let g be a simple lie algebra of exceptional type over an algebraically closed field k and let g be a simple linear algebraic group with lie algebra g for char k p 0 we present a complete listing of the g conjugacy classes of balanced toral elements of g as a result we also obtain the classification of conjugacy classes of balanced inner torsion automorphisms of g of order p when char k 0

characteristic polynomials of real symmetric random matrices the correlation functions of the random variables det x in which x is an hermitian n n random matrix are known to exhibit universal local statistics in the large n limit we study here the correlation of those same random variables for real symmetric matrices goe the derivation relies on an exact dual representation of the problem the k point functions are expressed in terms of finite integrals over quaternionic k k matrices however the control of the dyson limit in which the distance of the various parameters s is of the order of the mean spacing requires an integration over the symplectic group it is shown that a generalization of the itzykson zuber method holds for this problem but contrary to the unitary case the semi classical result requires a finite number of corrections to be exact we have also considered the problem of an external matrix source coupled to the random matrix and obtain explicit integral formulae which are useful for the analysis of the large n limit

petri nets event structures and domains part i abstract the general aim of this paper is to find a theory of concurrency combining the approaches of petri and scott and others in part i we introduce our formalisms to connect the abstract ideas of events and domains of information we show how casual nets induce certain kinds of domains where the information points are certain sets of events this allows translations between the languages of net theory and domain theory following the idea that events of causal nets are occurrences we generalise causal nets to occurrence nets by adding forwards conflict just as infinite flow charts unfold finite ones so transition nets can be unfolded into occurrence nets next we extend the above connections between nets and domains to these new nets event structures which are intermediate between nets and domains play an important part in all our work finally as an example of how concepts translate from one formalism to the other we show how petri s notion of confusion ties up with kahn and plotkin s concrete domains in part ii we shall continue the job of connecting up notions within net theory and the theory of domains in particular we shall examine the idea of states of computations

on the expressiveness of higher dimensional automata in this paper i compare the expressive power of several models of concurrency based on their ability to represent causal dependence to this end i translate these models in behaviour preserving ways into the model of higher dimensional automata hda which is the most expressive model under investigation in particular i propose four different translations of petri nets corresponding to the four different computational interpretations of nets found in the literature i also extend various equivalence relations for concurrent systems to hda these include the history preserving bisimulation which is the coarsest equivalence that fully respects branching time causality and their interplay as well as the st bisimulation a branching time respecting equivalence that takes causality into account to the extent that it is expressible by actions overlapping in time through their embeddings in hda it is now well defined whether members of different models of concurrency are equivalent

combinatorics of labelling in higher dimensional automata the main idea for interpreting concurrent processes as labelled precubical sets is that a given set of n actions running concurrently must be assembled to a labelled n cube in exactly one way the main ingredient is the non functorial construction called the labelled directed coskeleton it is defined as a subobject of the labelled coskeleton the latter coinciding in the unlabelled case with the right adjoint to the truncation functor this non functorial construction is necessary since the labelled coskeleton functor of the category of labelled precubical sets does not fulfil the above requirement we prove in this paper that it is possible to force the labelled coskeleton functor to be well behaved by working with labelled transverse symmetric precubical sets moreover we prove that this solution is the only one a transverse symmetric precubical set is a precubical set equipped with symmetry maps and with a new kind of degeneracy map called transverse degeneracy finally we also prove that the two settings are equivalent from a directed algebraic topological viewpoint to illustrate a new semantics of the calculus of communicating systems ccs equivalent to the old one is given

sub principal homomorphisms in positive characteristic let g be a reductive group over an algebraically closed field of characteristic p and let u g be a unipotent element of order p suppose that p is a good prime for g we show in this paper that there is a homomorphism sl 2 k g whose image contains u this result was first obtained by d testerman j algebra 1995 using case considerations for each type of simple group and using in some cases computer calculations with explicit representatives for the unipotent orbits the proof we give is free of case considerations except in its dependence on the bala carter theorem our construction of generalizes the construction of a principal homomorphism made by j p serre in invent math 1996 in particular is obtained by reduction modulo from a homomorphism of group schemes over a valuation ring in a number field this permits us to show moreover that the weight spaces of a maximal torus of sl 2 k on lie g are the same as in characteristic 0 the existence of a with this property was previously obtained again using case considerations by lawther and testerman memoirs ams 1999 and has been applied in some recent work of g seitz invent math 2000

the central limit theorem for local linear statistics in classical compact groups and related combinatorial identities author s soshnikov alexander abstract we discuss clt for the global and local linear statistics of random matrices from classical compact groups the main part of our proofs are certain combinatorial identities much in the spirit of works by kac and spohn

arbitrary unitarily invariant random matrix ensembles and supersymmetry we generalize the supersymmetry method in random matrix theory to ensembles which are unitarily invariant but otherwise arbitrary our exact approach extends a previous contribution in which we constructed a supersymmetric representation for the class of norm dependent random matrix ensembles here we derive a supersymmetric formulation under very general circumstances a reduced probability density and a projector are identified that map the probability density from ordinary to superspace furthermore it is demonstrated that setting up the theory in fourier superspace has considerable advantages general and exact expressions for the correlation functions are given we also show how the use of hyperbolic symmetry can be circumvented in the present context in which the nonlinear model is not used we construct exact supersymmetric integral representations of the correlation functions for arbitrary positions of the imaginary increments in the green s functions

new class of random matrix ensembles with multifractal eigenvectors department of physics university of florida gainesville florida 32611 february 1 2008 three recently suggested random matrix ensembles rme are linked together to represent a class of rme with multifrac tal eigenfunction statistics the generic form of the two levelcorrelation function for the case of weak and extremely strongmultifractality is suggested pacs numbers 71 25 s 72 15 rn 05 45 b

probability density of the determinant of a random hermitian matrix the probability density function for the determinant of a nn random hermitian matrix taken from the gaussian unitary ensemble is calculated it is found to be a meijer g function or a linear combination of two meijer g functions depending on the parity of n the integer moments of this probability density are also given

multifractal wavefunction at the localisation threshold the authors point out that the field theoretical description of the localisation transition exhibits anomalous scaling for the moments of the probability density at the localisation threshold this property hints at a multifractal structure of the wavefunction they attempt an analysis of the corresponding singularity distribution

transitions in spectral statistics we present long range statistical properties of a recently introduced unitary random matrix ensemble whose short range correlations were found to describe a transition from wigner to poisson type as a function of a single parameter we argue by evaluating the two level correlation function of a different solvable model that the transition is perhaps a quite general feature for a class of models our analytic results compare well with numerical studies on a variety of physical systems we discuss the possibility of observing experimentally the signature of such transitions

distribution of eigenvalues of certain matrix ensembles we investigate spectral properties of ensembles of n ifmmode times else texttimes fi n random matrices m defined by their probability distribution p m exp tr v m with a weekly confinement potential v m for which the moment problem mathrm ensuremath mu mathrm n ensuremath int mathrm x mathrm n exp v x dx is indeterminated the characteristic property of these ensembles is that the mean density of eigenvalues tends with increasing matrix dimension to be a continuous function contrary to the usual strong confinement cases where it grows indefinitely when n ensuremath rightarrow ensuremath infty we demonstrate that the standard asymptotic formulas for correlation functions are not applicable for weakly confinement ensembles and their asymptotic distribution of eigenvalues can deviate from the classical ones the model with v x ln 2 x ensuremath mathrm b is considered in detail it is shown that when ensuremath mathrm b ensuremath rightarrow ensuremath infty the unfolded eigenvalue distribution tends to a limit which is different from any standard random matrix ensembles but which is the same for all three symmetry classes unitary orthogonal and symplectic

weak morphisms of higher dimensional automata abstract we introduce weak morphisms of higher dimensional automata and use them to define preorder relations for hdas among which are homeomorphic abstraction and trace equivalent abstraction it is shown that homeomorphic abstraction is essentially always stronger than trace equivalent abstraction we also define the trace language of an hda and show that for a large class of hdas it is invariant under trace equivalent abstraction

magnetic fluctuations in the classical xy model the origin of an exponential tail in a complex system we study the probability density function for the fluctuations of the magnetic order parameter in the low temperature phase of the mathrm xy model of finite size in two dimensions this system is critical over the whole of the low temperature phase it is shown analytically and without recourse to the scaling hypothesis that in this case the distribution is non gaussian and of universal form independent of both system size and critical exponent ensuremath eta an exact expression for the generating function of the distribution is obtained which is transformed and compared with numerical data from high resolution molecular dynamics and monte carlo simulations the asymptotes of the distribution are calculated and found to be of exponential and double exponential form the calculated distribution is fitted to three standard functions a generalization of gumbel s first asymptote distribution from the theory of extremal statistics a generalized log normal distribution and a ensuremath chi 2 distribution the calculation is extended to general dimension and an exponential tail is found in all dimensions less than 4 despite the fact that critical fluctuations are limited to d 2 these results are discussed in the light of similar behavior observed in models of interface growth and for dissipative systems driven into a nonequilibrium steady state

norm dependent random matrix ensembles in external field and supersymmetry the class of norm dependent random matrix ensembles is studied in the presence of an external field the probability density of the ensemble depends on the trace of the squared random matrices but is otherwise arbitrary an exact mapping to superspace is performed a transformation formula is derived which gives the probability density in superspace as a single integral over the probability density in ordinary space this is done for orthogonal unitary and symplectic symmetries in the case of unitary symmetry some explicit results for the correlation functions are derived

probability density of determinants of random matrices in this brief paper the probability density of a random real complex and quaternion determinant is rederived using the singular values the behaviour of suitably rescaled random determinants is studied in the limit of infinite order of the matrices

joint probability densities of level spacing ratios in random matrices we calculate analytically for finite size matrices joint probability densities of ratios of level spacings in ensembles of random matrices characterized by their associated confining potential we focus on the ratios of two spacings between three consecutive real eigenvalues as well as certain generalizations such as the overlapping ratios the resulting formulas are further analyzed in detail in two specific cases the hermite and the laguerre cases for which we offer explicit calculations for small n the analytical results are in excellent agreement with numerical simulations of usual random matrix ensembles and with the level statistics of a quantum many body lattice model and zeros of the riemann zeta function

nilpotent orbits in good characteristic and the kempf rousseau theory let g be a connected reductive algebraic group over an algebraically closed field k of characteristic p 0 image and suppose that p is a good prime for the root system of g in this paper we give a fairly short conceptual proof of pommerening s theorem pommerening j algebra 49 1977 525 536 j algebra 65 1980 373 398 which states that any nilpotent element in image is richardson in a distinguished parabolic subalgebra of the lie algebra of a levi subgroup of g as a by product we obtain a short noncomputational proof of the existence theorem for good transverse slices to the nilpotent g orbits in image for earlier proofs of this theorem see kawanaka invent math 84 1986 575 616 premet trans amer math soc 347 1995 2961 2988 spaltenstein j fac sci univ tokyo sect ia math 31 1984 283 286 we extend recent results of sommers internal math res notices 11 1998 539 562 to reductive lie algebras of good characteristic thus providing a satisfactory approach to computing the component groups of the centralisers of nilpotent elements in image and unipotent elements in g earlier computations of these groups in positive characteristics relied mostly on work of mizuno j fac sci univ tokyo sect ia math 24 1977 525 563 tokyo j math 3 1980 391 459 our approach is based on the theory of optimal parabolic subgroups for g unstable vectors also known as the kempf rousseau theory which provides a good substitute for the image theory prominent in the characteristic zero case

superstatistical generalizations of wishart laguerre ensembles of random matrices using beck and cohen s superstatistics we introduce in a systematic way a family of generalised wishart laguerre ensembles of random matrices with dyson index 1 2 and 4 the entries of the data matrix are gaussian random variables whose variancesfluctuate from one sample to another according to a certain probability density f and a single deformation parameter three superstatistical classes for f are usually considered 2 inverse 2 and log normal distributions while the first class already considered by two of the authors leads to a power law decay of the spectral density we here introduce and solve exactly a superposition of wishart laguerre ensembles with inverse 2 distribution the corresponding macroscopic spectral density is given by a deformation of the semi circle and marcenko pastur laws on a non compact support with exponential tails after discussing in detail the validity of wigner s surmise in the wishart laguerre class we introduce a generalised dependent surmise with stretched exponential tails which well approximates the individual level spacing distribution in the bulk the analytical results are in excellent agreement with numerical simulations to illustrate our findings we compare the 2 and inverse 2 class to empirical data from financial covariance matrices

limiting distributions for a polynuclear growth model with external sources the purpose of this paper is to investigate the limiting distribution functions for a polynuclear growth model with two external sources which was considered by prahofer and spohn depending on the strength of the sources the limiting distribution functions are either the tracy widom functions of random matrix theory or a new explicit function which has the special property that its mean is zero moreover we obtain transition functions between pairs of the above distribution functions in suitably scaled limits there are also similar results for a discrete totally asymmetric exclusion process