probability of all eigenvalues real for products of standard gaussian matrices with xi independent n n standard gaussian random matrices the probability that all eigenvalues are real for the matrix product pm xmxm 1 x1 is expressed in terms of an n 2 n 2 n even and n 1 2 n 1 2 n odd determinant the entries of the determinant are certain meijer g functions in the case m 2 high precision computation indicates that the entries are rational multiples of 2 with the denominator a power of 2 and that to leading order in n decays as we are able to show that for general m and large n with an explicit bm an analytic demonstration that as m is given

polynomial time 1 turing reductions from ph to p in this paper we investigate relative complexity between p and other classes of functions our particular interest is to compare p with ph and with pfh by using polynomial time reducibility and to demonstrate that a weaker notion of polynomial time reducibility is sufficiently powerful for reducing ph functions to p functions our main result is stated as follows every function in ph is polynomial time 1 turing reducible to some function in p that is ph pf p 1 some consequences of this result are as follows every function in pfh is polynomial time 1 turing reducible to some function in p if pf p 1 ph then ph collapses to a finite level furthermore if either p pfh or pfh p then ph collapses to a finite level we also give an affirmative answer to an open question posed by valiant 1979 and we show a generalized result about p rankability by hemachandra 1987

some universal properties for restricted trace gaussian orthogonal unitary and symplectic ensembles consider fixed and bounded trace gaussian orthogonal unitary and symplectic ensembles closely related to gaussian ensembles without any constraint for three restricted trace gaussian ensembles we prove universal limits of correlation functions at zero and at the edge of the spectrum edge our argument also applies to restricted trace ensembles with monomial potentials in addition by using the universal result in the bulk for fixed trace gaussian unitary ensemble which has been obtained by gotze and gordin we also prove the universal limits of correlation functions everywhere in the bulk for bounded trace gaussian unitary ensemble

matrices coupled in a chain ii spacing functions for the eigenvalues of p complex hermitian matrices coupled in a chain we give a method of calculating the spacing functions this is a generalization of the one matrix case which has been known for a long time

distribution of bipartite entanglement for random pure states we calculate analytic expressions for the distribution of bipartite entanglement for pure random quantum states all moments of the purity distribution are derived and an asymptotic expansion for the distribution itself is deduced an approximate expression for moments and distribution of meyer wallach entanglement for random pure states is then obtained

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

the computational complexity of maximization and integration on montre que ces deux problemes de la theorie de la complexite de calcul aussi bien que les autres resolvant la maximisation et l integration sont equivalents aux problemes de complexite de calcul discret naturel dont la plupart ont deja beaucoup ete etudies

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

enumerative counting is hard abstract an n variable boolean formula may have anywhere from 0 to 2n satisfying assignments can a polynomial time machine given such a formula reduce this exponential number of possibilities to a small number of possibilities we call such a machine an enumerator and prove that if there is a good polynomial time enumerator for p i e one where for every boolean formula f the small set has at most o f 1 e numbers then p np p p and probabilistic polynomial time equals polynomial time furthermore we show that p polynomial time turing reduces to enumerating p

complexity classes of optimization functions in this paper complexity classes of functions defined via taking maxima or minima cf the work of krentel or taking middle elements cf the work of toda are examined a number of axioms for a class to be a so called p founded class of optimization functions are given it is shown that many natural function classes fulfill these axioms then these classes are examined concerning their relationship to complexity classes of sets to this end complexity preserving operators s and f for encoding function classes into set classes and vice versa are introduced it is shown how these operators translate closure properties from one class to another how they relate operators on classes of functions and classes of sets and how they encode classes of maximum minimum or median functions into well studied classes of sets the fixpoints of the compositional operator f s are examined and shown to be exactly those function classes closed under binary search let f1 and f2 be two such fixpoints and k1 and k2 be their corresponding classes of sets i e their images under the operator s then f1 f2 if and only if k1 k2 and f1 f2 if and only if k1 k2 a number of natural classes of functions are shown to be such fixpoints thus we build hierachies of function classes with the same inclusional relationships as the polynomial time hierarchy of sets and the counting hierarchy of sets and we prove many interesting structural properties of these hierarchies

on some natural complete operators abstract an operator is a mapping from integer functions to integer functions it is known from a result by baker gill and solovay 1975 that there exists an operator computable in polynomial time by a nondeterministic oracle turing machine called an np operator but not computable in polynomial time by any deterministic oracle turing machine we investigate several natural operators which share similar properties we use the concept of completeness to give a precise classification of the complexity of these operators for example the question of finding maximum values of polynomial time computable functions can be formulated as an operator complete for the class of np operators

universal fluctuations and extreme value statistics we study the effect of long range algebraic correlations on extreme value statistics and demonstrate that correlations can produce a limit distribution which is indistinguishable from the ubiquitous bramwell holdsworth pinton distribution we also consider the square width fluctuations of the avalanche signal we find as recently predicted by antal et al for logarithmic correlated 1 f signals that these fluctuations follow the fisher tippett gumbel distribution from uncorrelated extreme value statistics

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

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

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

on the universality of the probability distribution of the product b 1x of random matrices consider random matrices a of dimension m x m n drawn from an ensemble with probability density f tr aa with f x a given appropriate function break a b x into an m x m block b and the complementary m x n block x and define the random matrix z b 1 x we calculate the probability density function p z of the random matrix z and find that it is a universal function independent of f x the universal probability distribution p z is a spherically symmetric matrix variate t distribution universality of p z is essentially a consequence of rotational invariance of the probability ensembles we study as an application we study the distribution of solutions of systems of linear equations with random coefficients and extend a classic result due to girko

recursion theoretic characterizations of complexity classes of counting functions abstract there has been a great effort in giving machine independent algebraic characterizations of complexity classes especially of functions astonishingly no satisfactory characterization of the prominent class p is known up to now here we characterize p as the closure of a set of simple arithmetical functions under summation and weak product based on that result the hierarchy of counting functions which is the closure of p under substitution is characterized remarkably without using the operator of substitution since we can show that in the context of this hierarchy the operation of modified subtraction is as powerful as substitution this leads us to a number of consequences concerning closure of p under certain arithmetical operations analogous results are achieved for the class gap p which is the closure of p under subtraction

exact density of states for finite gaussian random matrix ensembles via supersymmetry we calculate the exact density of states dos for the three classical and two non classical random matrix ensembles for finite matrix size n using supersymmetric integrals the 1 n expansion yields already in lowest order good approximations to the exact result even for small values of n 5 we conjecture a connection between the n dependence of the oscillating part of the dos and the short distance behaviour of the two level correlation function

calculation of a certain determinant the n n determinant det a j i b j i is evaluated this completes the calculation of the mellin transform of the probability density of the determinant of a random quaternion self dual matrix taken from the gaussian symplectic ensemble the inverse mellin transform then gives the later probability density itself

the spectral parabolic equation and three dimensional backscattering problems involving dependence of the acoustic parameters in depth and one of the horizontal cartesian coordinates are solved using the cross range wave number spectrum and the parabolic equation pe method the self starter and the two way pe are used to solve the two dimensional wave equation that is obtained after transforming to wave number space although the spectral pe solution breaks down for energy that propagates nearly parallel to the cross range axis it is accurate in other directions for both outgoing and backscattered energy the spectral pe is practical for solving large scale reverberation problems in three dimensions

generalized extreme value statistics and sum of correlated variables we show that generalized extreme value statistics the statistics of the kth largest value among a large set of random variables can be mapped onto a problem of random sums this allows us to identify classes of non identical and generally correlated random variables with a sum distributed according to one of the three k dependent asymptotic distributions of extreme value statistics namely the gumbel frechet and weibull distributions these classes as well as the limit distributions are naturally extended to real values of k thus providing a clear interpretation to the onset of gumbel distributions with non integer index k in the statistics of global observables this is one of the very few known generalizations of the central limit theorem to non independent random variables finally in the context of a simple physical model we relate the index k to the ratio of the correlation length to the system size which remains finite in strongly correlated systems

some observations on the connection between counting and recursion abstract based on valiant s class p of all functions counting the number of accepting computations of nondeterministic polynomial time turing machines the polynomial time hierarchy of counting functions is introduced the class phcf of all functions of this hierarchy and some of its subclasses are characterized by recursion theoretic means it turns out that from the recursion theoretic point of view phcf is an analogue to kalmar s class e of elementary functions to the class pspace of polynomial space computable functions as well as to the class p of polynomial time computable functions

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

three dimensional propagation modeling in shallow water much interest exists in problems of shallow water propagation where three dimensional spatially dependent waveguide effects are sometimes very large in these cases the three dimensional dependence of the acoustic field can be accounted for in a variety of ways other than solving the often intractable fully coupled problem there exist several pseudo three dimensional models that are based on the parabolic equation method e g n 2d pe j s perkins and r n bear j acoust soc am 72 515 522 1982 the adiabatic pe m d collins j acoust soc am 93 2269 2278 1993 the coupled mode pe a t abawi et al j acoust soc am 102 233 238 1997 and the spectral pe g j orris and m d collins j acoust soc am 96 3499 3503 1994 yet because these models are extensions of efficient two dimensional models they have explicit and implicit limitations on their applicability some of the advantages and limitations of these pseudo three dimensional models will be demonstrated through their application to data collected during the swellex 96 and rds 1 matched field experiments

two remarks on the power of counting the relationship between the polynomial hierarchy and valiant s class p is at present unknown we show that some low portions of the polynomial hierarchy namely deterministic polynomial algorithms using an np oracle at most a logarithmic number of times can be simulated by one p computation we also show that the class of problems solvable by polynomial time nondeterministic turing machines which accept whenever there is an odd number of accepting computations is idempotent that is closed under usage of oracles from the same class

distribution of the determinant of a random real symmetric matrix from the gaussian orthogonal ensemble random matrix theory finds considerable use in various branches of physics notably in nuclear physics quantum chaology and for investigating hamiltonians of disordered and strongly interacting quantum systems 1 3 and references therein among the ensembles of n3n random matrices hn three gaussian ensembles have been studied extensively 1 3 and are still being investigated their probability densities are proportional to exp 2tr hn where tr means trace matrices are real symmetric for the gaussian orthogonal ensemble goe hermitian for the gaussian unitary ensemble gue and quaternion self dual for the gaussian symplectic ensemble many characteristics both exact at finite n and asymptotic at large n of the distributions of eigenvalues of n3n random matrices of various symmetries are known for the gaussian ensembles 1 3 the properties of eigenvalues of random matrix ensembles rme s can be interpreted in two dimensions 2d from the equilibrium characteristics of a gas of n identical point charges on a line 1 10 often referred to as a log gas 9 10 which interact via a logarithmic coulomb potential and are confined by an external potential the external potential is harmonic in the case of gaussian ensembles as can be seen by rewriting eq 1 below as a boltzmann factor at a temperature 1 b that depends only on the symmetry of the considered ensemble with b51 2 and 4 for the orthogonal unitary and symplectic ensembles respectively 1 3 the problems of the distribution and associated fluctuations of linear statistics in rme s 1 10 and references therein are of interest for instance in the study of conductance fluctuations in mesoscopic conductors the linear statistics problem deals with physical quantities f which are given by sums f5 k51 n f l k over the eigenvalues l k of a random matrix where f x may depend nonlinearly on x general arguments have been used to predict that the distribution of any linear statistic is gaussian and independent of the confining potential in the scaled asymptotic limit n provided its variance is finite 5 10 the potential fluctuations have been investigated in 2d and 3d classical charged systems 11 12 and more recently for circular ensembles by baker and forrester who further studied the dipole moment statistic 9 within the log gas interpretation the statistic of the potential at the origin is obtained for f x 52lnuxu that is for f5 2lnudu it may thus be derived directly from the distribution of the determinant d of hn we notice further that the partition functions of log gases whose confining harmonic potential includes a supplementary logarithmic contribution are by products of the derivation of the determinant distributions of gaussian ensembles see eqs 10 12 19 and 26 for b51 the rme s associated with the latter log gases belong to the family of generalized gaussian ensembles 13 14

evaluation of a certain q determinant abstract a q analogue of mehta wang s determinant is introduced and evaluated in the case when q 1 this determinant contains a factor represented by using a special case of the meixner pollaczek polynomial a recurrence relation for the factor is derived in the case when 0 q

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

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

nearest neighbour spacing distributions of the β hermite ensemble of random matrices the evolution with of the distributions of the spacing s between nearest neighbour levels of unfolded spectra of random matrices from the hermite ensemble he is investigated by monte carlo simulations the random matrices from the he are real symmetric and tridiagonal where which can take any positive value is the reciprocal of the temperature in the classical electrostatic interpretation of eigenvalues the distribution of eigenvalues coincide with those of the three classical gaussian ensembles for 1 2 4 the use of the he ensemble results in an incomparable speed up and efficiency of numerical simulations of all spectral characteristics of large random matrices generalized gamma distributions are shown to be excellent approximations of the nearest neighbor spacing nns distributions for any while being still simple they account both for the level repulsion in s when s 0 and for the whole shape of the nns distributions in the range of s which is accessible to experiment or to most numerical simulations the exact nns distribution of the goe 1 is in particular significantly better described by a generalized gamma distribution than it is by the wigner surmise while the best generalized gamma approximation coincides essentially with the wigner surmise for 2 they describe too the evolution of the level repulsion between that of a poisson distribution and that of a goe distribution when increases from 0 to 1 the distribution of ln s related to the electrostatic interaction energy between neighbouring charges is accordingly well approximated by a generalized gumbel distribution for any 0 the distributions of the minimum nn spacing between eigenvalues of matrices from the he obtained both from as calculated eigenvalues and from unfolded eigenvalues are brody distributions which are classically used to characterize the spectral fluctuations of various physical systems

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

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

the δn statistic for the β hermite ensemble abstract the fluctuation n of the n th unfolded eigenvalue was recently characterized for the classical gaussian ensembles of n n random matrices goe gue gse it is investigated here for the hermite ensemble as a function of the reciprocal of the temperature by monte carlo simulations the ensemble averaged fluctuation n 2 and the autocorrelation function vary logarithmically with n for any 0 1 n n the simple logarithmic behavior of the higher order moments of n reported in the literature for the goe 1 and the gue 2 holds for any 0 and is accounted for by gaussian distributions whose variances depend linearly on ln n

a three dimensional propagation algorithm using finite azimuthal aperture the use of three dimensional 3 d propagation models is becoming increasingly common in current underwater acoustics applications such models typically treat the propagation as a function of range depth and bearing consistent with previous multibearing two dimensional n 2 d models however to obtain accurate 3 d solutions at long ranges many bearings must be computed in order to maintain the necessary cross range resolution between bearings in this paper a 3 d parabolic equation pe model is developed with a marching algorithm based on the split step fourier technique in both depth and azimuthal bearing the algorithm includes a scheme to compute the solution for only a finite azimuthal aperture rather than the full 360 of bearing the success of this algorithm allows for increased cross range resolution at long range without increasing the number of bearings needed in the calculation results of this technique are compared to results from a suggested benchmark case

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

a central limit theorem for the determinant of a wigner matrix abstract we establish a central limit theorem for the log determinant log det m n of a wigner matrix m n under the assumption of four matching moments with either the gue or goe ensemble more specifically we show that this log determinant is asymptotically distributed like n log n 1 2 log n 1 2 log n r when one matches moments with gue and n log n 1 4 log n 1 4 log n r when one matches moments with goe

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

a decisive characterization of bpp the complexity class bpp defined by gill contains problems that can be solved in polynomial time with bounded error probability a new and simple characterization of bpp is given it is shown that a language l is in bpp iff x l y zp x y z x l y z p x y z for a polynomial time predicate p and for y z poly x the formula yp y with the random quantifier means that the probability pr y p y for a fixed this characterization allows a simple proof that bpp zpp np which strengthens the result of lautemann inform process lett 17 1983 215 217 sipser in proceedings 15th annu acm sympos theory of comput 1983 pp 330 335 that bpp 2 p 2 p several other results about probabilistic classes can be proved using similar techniques e g np r zpp np and 2 p bpp 2 p

stanley s zrank conjecture on skew partitions we present an affirmative answer to stanley s zrank conjecture namely the zrank and the rank are equal for any skew partition we show that certain classes of restricted cauchy matrices are nonsingular and furthermore the signs are determined by the number of zero entries we also give a characterization of the rank in terms of the giambelli type matrices of the corresponding skew schur functions our approach also applies to the factorial cauchy matrices and the inverse binomial coefficient matrices

developments in random matrix theory in this introduction to the journal of physics a special issue on random matrix theory we give a review of the main historical developments in random matrix theory a short summary of the papers that appear in this special issue is also given

nonlinear random matrix statistics symmetric functions and hyperdeterminants nonlinear statistics i e statistics of permanents on the eigenvalues of invariant random matrix models are considered for the three dyson s symmetry classes 1 2 4 general formulas in terms of hyperdeterminants are found for 2 for specific cases and all s more computationally efficient results are obtained based on symmetric functions expansions as an application we consider the case of quantum transport in chaotic cavities extending results from savin et al 2008 phys rev b 77 125332

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

moments of the characteristic polynomial in the three ensembles of random matrices moments of the characteristic polynomial of a random matrix taken from any of the three ensembles orthogonal unitary or symplectic are given either as a determinant or a pfaffian or as a sum of determinants for gaussian ensembles on comparing two expressions of the same moment one obtains two remarkable identities one between an n n determinant and an m m determinant and another between the pfaffian of a 2n 2n anti symmetric matrix and a sum of m m determinants

block orthogonal polynomials ii hermite and laguerre standard block orthogonal polynomials the standard block orthogonal sbo polynomials pi n x 0 i n are real polynomials of degree n which are orthogonal with respect to a first euclidean scalar product to polynomials of degree less than i in addition they are mutually orthogonal with respect to a second euclidean scalar product applying the general results obtained in a previous paper we determine and investigate these polynomials when the first scalar product corresponds to hermite resp laguerre polynomials these new sets of polynomials which we call hermite resp laguerre sbo polynomials provide a basis of functional spaces well suited for some applications requiring to take into account special linear constraints which can be recast into an euclidean orthogonality relation

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

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

compact support probability distributions in random matrix theory we consider a generalization of the fixed and bounded trace ensembles introduced by bronk and rosenzweig up to an arbitrary polynomial potential in the large n limit we prove that the two are equivalent and that their eigenvalue distribution coincides with that of the canonical ensemble with measure exp ntr v m the mapping of the corresponding phase boundaries is illuminated in an explicit example in the case of a gaussian potential we are able to derive exact expressions for the one and two point correlator for finite n having finite support

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