derived equivalences of canonical covers of hyperelliptic and enriques surfaces in positive characteristic we prove that any fourier mukai partner of an abelian surface over an algebraically closed field of positive characteristic is isomorphic to a moduli space of gieseker stable sheaves we apply this

dynamic scaling of the width distribution in edwards wilkinson type models of interface dynamics edwards wilkinson type models are studied in 1 1 dimensions and the time dependent distribution mathit p mathit l mathit w 2 t of the square of the width of an interface mathit w 2 is calculated for systems of size l we find that using a flat interface as an initial condition mathit p mathit l mathit w 2 t can be calculated exactly and it obeys scaling in the form mathit w 2 mathrm mathrm ensuremath infty mathit p mathit l mathit w 2 t ensuremath phi mathit w 2 mathit w 2 mathrm mathrm ensuremath infty t mathrm l 2 where mathit w 2 mathrm mathrm ensuremath infty is the stationary value of mathit w 2 for more complicated initial states scaling is observed only in the large time limit and the scaling function depends on the initial amplitude of the longest wavelength mode the short time limit is also interesting since mathit p mathit l mathit w 2 t is found to closely approximate the log normal distribution these results are confirmed by monte carlo simulations on a single step solid on solid type model roof top model of surface evolution textcopyright 1996 the american physical society

phase correlation of foreign exchange time series abstract correlation of foreign exchange rates in currency markets is investigated based on the empirical data of usd dem and usd jpy exchange rates for a period from february 1 1986 to december 31 1996 the return of exchange time series is first decomposed into a number of intrinsic mode functions imfs by the empirical mode decomposition method the instantaneous phases of the resultant imfs calculated by the hilbert transform are then used to characterize the behaviors of pricing transmissions and the correlation is probed by measuring the phase differences between two imfs in the same order from the distribution of phase differences our results show explicitly that the correlations are stronger in daily time scale than in longer time scales the demonstration for the correlations in periods of 1986 1989 and 1990 1993 indicates two exchange rates in the former period were more correlated than in the latter period the result is consistent with the observations from the cross correlation calculation

numerical convergence of the block maxima approach to the generalized extreme value distribution in this paper we perform an analytical and numerical study of extreme value distributions in discrete dynamical systems in this setting recent works have shown how to get a statistics of extremes in agreement with the classical extreme value theory we pursue these investigations by giving analytical expressions of extreme value distribution parameters for maps that have an absolutely continuous invariant measure we compare these analytical results with numerical experiments in which we study the convergence to limiting distributions using the so called block maxima approach pointing out in which cases we obtain robust estimation of parameters in regular maps for which mixing properties do not hold we show that the fitting procedure to the classical extreme value distribution fails as expected however we obtain an empirical distribution that can be explained starting from a different observable function for which nicolis et al phys rev lett 97 21 210602 2006 have found analytical results

relevance of soft modes for order parameter fluctuations in the two dimensional xy model we analyse the spin wave approximation for the two dimensional xy model directly in reciprocal space in this limit the model is diagonal and the normal modes are statistically independent despite this simplicity non trivial critical properties are observed and exploited we confirm that the observed asymmetry for the probability density function for order parameter fluctuations comes from the divergence of the mode amplitudes across the brillouin zone we show that the asymmetry is a many body effect despite the importance played by the zone centre the precise form of the function is dependent on the details of the microscopic generating function for the modes we discuss the relevance of this to more complex or non equilibrium systems where similar distributions are observed

the coupling analysis between stock market indices based on permutation measures abstract many information theoretic methods have been proposed for analyzing the coupling dependence between time series and it is significant to quantify the correlation relationship between financial sequences since the financial market is a complex evolved dynamic system recently we developed a new permutation based entropy called cross permutation entropy cpe to detect the coupling structures between two synchronous time series in this paper we extend the cpe method to weighted cross permutation entropy wcpe to address some of cpe s limitations mainly its inability to differentiate between distinct patterns of a certain motif and the sensitivity of patterns close to the noise floor it shows more stable and reliable results than cpe does when applied it to spiky data and ar 1 processes besides we adapt the cpe method to infer the complexity of short length time series by freely changing the time delay and test it with gaussian random series and random walks the modified method shows the advantages in reducing deviations of entropy estimation compared with the conventional one finally the weighted cross permutation entropy of eight important stock indices from the world financial markets is investigated and some useful and interesting empirical results are obtained

multifractality in the random parameter model for multivariate time series abstract the random parameter model was proposed to explain the structure of the covariance matrix in problems where most but not all of the eigenvalues of the covariance matrix can be explained by random matrix theory in this article we explore the scaling properties of the model as observed in the multifractal structure of the simulated time series we use the wavelet transform modulus maxima technique to obtain the multifractal spectrum dependence with the parameters of the model the model shows a scaling structure compatible with the stylized facts for a reasonable choice of the parameter values

multivariate markov chain modeling for stock markets abstract we study a multivariate markov chain model as a stochastic model of the price changes of portfolios in the framework of the mean field approximation the time series of price changes are coded into the sequences of up and down spins according to their signs we start with the discussion for small portfolios consisting of two stock issues the generalization of our model to arbitrary size of portfolio is constructed by a recurrence relation the resultant form of the joint probability of the stationary state coincides with gibbs measure assigned to each configuration of spin glass model through the analysis of actual portfolios it has been shown that the synchronization of the direction of the price changes is well described by the model

width distribution for random walk interfaces roughening of a one dimensional interface is studied under the assumption that the interface configurations are continuous periodic random walks the distribution of the square of the width of interface mathit w 2 is found to scale as p mathit w 2 mathit w 2 mathrm mathrm ensuremath 1 ensuremath phi mathit w 2 mathit w 2 where mathit w 2 is the average of mathit w 2 we calculate the scaling function ensuremath phi x exactly and compare it both to exact enumerations for a discrete slope surface evolution model and to ensuremath phi s obtained in monte carlo simulations of equilibrium and driven interfaces of chemically reacting systems

dynamic scaling approach to study time series fluctuations we propose an approach for properly analyzing stochastic time series by mapping the dynamics of time series fluctuations onto a suitable nonequilibrium surface growth problem in this framework the fluctuation sampling time interval plays the role of time variable whereas the physical time is treated as the analog of spatial variable in this way we found that the fluctuations of many real world time series satisfy the analog of the family viscek dynamic scaling ansatz this finding permits us to use the powerful tools of kinetic roughening theory to classify model and forecast the fluctuations of real world time series

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

multifractal analysis and topological properties of a new family of weighted koch networks abstract weighted complex networks especially scale free networks which characterize real life systems better than non weighted networks have attracted considerable interest in recent years studies on the multifractality of weighted complex networks are still to be undertaken in this paper inspired by the concepts of koch networks and koch island we propose a new family of weighted koch networks and investigate their multifractal behavior and topological properties we find some key topological properties of the new networks their vertex cumulative strength has a power law distribution there is a power law relationship between their topological degree and weight strength the networks have a high weighted clustering coefficient of 0 41004 which is independent of the scaling factor c in the limit of large generation t the second smallest eigenvalue 2 and the maximum eigenvalue n are approximated by quartic polynomials of the scaling factor c for the general laplacian operator while 2 is approximately a quartic polynomial of c and n 1 5 for the normalized laplacian operator then we find that weighted koch networks are both fractal and multifractal their fractal dimension is influenced by the scaling factor c we also apply these analyses to six real world networks and find that the multifractality in three of them are strong

extreme value statistics in records with long term persistence many natural records exhibit long term correlations characterized by a power law decay of the autocorrelation function c s ensuremath sim s ensuremath ensuremath gamma with time lag s and correlation exponent 0l ensuremath gamma l1 we study how the presence of such correlations affects the statistics of the extreme events i e the maximum values of the signal within time segments of the fixed duration r we find numerically that i the integrated distribution function of the maxima converges to a gumbel distribution for large r similar to uncorrelated signals ii the deviations for finite r depend on the initial distribution of the records and on their correlation properties iii the maxima series exhibit long term correlations similar to those of the original data and most notably iv the maxima distribution as well as the mean maxima significantly depend on the history in particular on the previous maximum the last item implies that conditional mean maxima and conditional maxima distributions with the value of the previous maximum as condition should be considered for an improved extreme event prediction we provide indications that this dependence of the mean maxima on the previous maximum occurs also in observational long term correlated records

modelling gap size distribution of parked cars using random matrix theory abstract we apply the random matrix theory to the car parking problem for this purpose we adopt a coulomb gas model that associates the coordinates of the gas particles with the eigenvalues of a random matrix the nature of interaction between the particles is consistent with the tendency of the drivers to park their cars near to each other and in the same time keep a distance sufficient for manoeuvring we show that the recently measured gap size distribution of parked cars in a number of roads in central london is well represented by the spacing distribution of a gaussian unitary ensemble

finite size effects and intermittency in a simple aging system we study the intermittent dynamics and the fluctuations of the dynamic correlation function of a simple aging system given its size l and its coherence length xi the system can be divided into n independent subsystems where n frac l xi d and d is the dimension of space each of them is considered as an aging subsystem which evolves according to an activated dynamics between energy levels we compute analytically the distribution of trapping times for the global system which can take power law stretched exponential or exponential forms according to the values of n and the regime of times considered an effective number of subsystems at age t w n eff t w can be defined which decreases as t w increases as well as an effective coherence length xi t w sim t w 1 mu d where mu 1 characterizes the trapping times distribution of a single subsystem we also compute the probability distribution functions of the time intervals between large decorrelations which exhibit different power law behaviours as t w increases or n decreases and which should be accessible experimentally finally we calculate the probability distribution function of the two time correlator we show that in a phenomenological approach where n is replaced by the effective number of subsystems n eff t w the same qualitative behaviour as in experiments and simulations of several glassy systems can be obtained

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

volatility and correlation based systemic risk measures in the us market this paper deals with the problem of how to use simple systemic risk measures to assess portfolio risk characteristics using three simple examples taken from previous literature one based on raw and partial correlations another based on the eigenvalue decomposition of the covariance matrix and the last one based on an eigenvalue entropy a granger causation analysis revealed some of them are not always a good measure of risk in the sp therefore in the sense of risk as volatility the indicators are not always suitable nevertheless their results towards returns are similar to previous works that accept them a deeper analysis has shown that any symmetric measure based on eigenvalue decomposition of correlation matrices however is not useful as a measure of correlation risk the empirical counterpart analysis of this proposition stated that negative correlations are usually small and therefore do not heavily distort the behavior of the indicator

scaling analysis of trends using dfa we study two aspects of the detrended uctuation analysis dfa method namely the scaling behavior of the leading terms of the best 0t polynomials and the detection of trends we show analytically and numerically that the standard deviation of the leading terms of the best 0t polynomials used in dfa displays scaling behavior furthermore we demonstrate that the distribution of these terms can be used to reveal the presence of trends in the data we also argue that this distribution can be used as a sensitive tool for identifying weak trends c 2001 elsevier science b v all rights reserved

semistable sheaves in positive characteristic we prove maruyama s conjecture on the boundedness of slope semistable sheaves on a projective variety defined over a noetherian ring our approach also gives a new proof of the boundedness for varieties defined over a characteristic zero field this result implies that in mixed characteristic the moduli spaces of gieseker semistable sheaves are projective schemes of finite type the proof uses a new inequality bounding slopes of the restriction of a sheaf to a hypersurface in terms of its slope and the discriminant this inequality also leads to effective restriction theorems in all characteristics improving earlier results in characteristic zero

complex surfaces with equivalent derived categories we examine the extent to which a smooth minimal complex projective surface x is determined by its derived category of coherent sheaves dx to do this we find for each such surface x the set of surfaces y for which there exists a fourier mukai transform d y to d x

tate s conjecture for k3 surfaces of finite height this paper extends the proof 16 of the tate conjecture for ordinary k3 surfaces over a finite field to the more general case of all k3 s of finite height as in 16 our method is to find a lifting of the k3 to characteristic zero with sufficiently many hodge cycles in the ordinary case the so called canonical lifting of deligne and illusie 7 did the job and a study of the galois action on p adic etale cohomology revealed the hodge cycles here we use more general quasi canonical liftings and the action of the crystalline weil group on

detecting synchronization clusters in multivariate time series via coarse graining of markov chains synchronization cluster analysis is an approach to the detection of underlying structures in data sets of multivariate time series starting from a matrix r of bivariate synchronization indices a previous method utilized the eigenvectors of r for cluster identification analogous to several recent attempts at group identification using eigenvectors of the correlation matrix all of these approaches assumed a one to one correspondence of dominant eigenvectors and clusters which has however been shown to be wrong in important cases we clarify the usefulness of eigenvalue decomposition for synchronization cluster analysis by translating the problem into the language of stochastic processes and derive an enhanced clustering method harnessing recent insights from the coarse graining of finite state markov processes we illustrate the operation of our method using a simulated system of coupled lorenz oscillators and we demonstrate its superior performance over the previous approach finally we investigate the question of robustness of the algorithm against small sample size which is important with regard to field applications

temperature dependence of universal fluctuations in the two dimensional harmonic xy model we compute exact analytical expressions for the skewness and kurtosis in the two dimensional harmonic xy model these quantities correspond to the third and fourth normalized moments of the probability density function pdf of the magnetization of the model from their behavior we conclude that they depend explicitly on the system temperature even in the thermodynamic limit and hence the pdf itself must depend on it our results correct the hypothesis called universal fluctuations they confirm and extend previous results which showed a t dependence of the pdf including perturbative expansions within the xy model up to first order in temperature

applications of extreme value statistics in physics we present a descriptive review of physical problems dealing with extreme values in several fields of physics we consider different physical situations involving random variables that are correlated or not and study the statistics of extremal variables which is relevant for situations where height fluctuations catastrophic events such as material failure or power outrage occur we describe the general theory and relate the cumulative limit distributions that can be accessible in experiments to microscopic models in many cases however the random variables are correlated in interface problems for example and the characteristics of the interaction are revealed in the asymptotic behavior of the limit distribution

temperature dependent fluctuations in the two dimensional xy model we present a detailed investigation of the probability density function pdf of order parameter fluctuations in the finite two dimensional xy 2dxy model in the low temperature critical phase of this model the pdf approaches a universal non gaussian limit distribution in the limit t 0 our analysis resolves the question of temperature dependence of the pdf in this regime for which conflicting results have been reported we show analytically that a weak temperature dependence results from the inclusion of multiple loop graphs in a previously derived graphical expansion this is confirmed by numerical simulations on two controlled approximations to the 2dxy model the harmonic and harmonic xy models the harmonic model has no kosterlitz thouless berezinskii ktb transition and the pdf becomes progressively less skewed with increasing temperature until it closely approximates a gaussian function above t approximate to 4 pi near to that temperature we find some evidence of a phase transition although our observations appear to exclude a thermodynamic singularity

universality classes for extreme value statistics the equilibrium low temperature physics of disordered systems is governed by the statistics of extremely low energy states it is thus relevant to discuss the possible universality classes for extreme value statistics we compare the usual probabilistic classification to the results of the replica approach we show in detail for several problems including the random energy model and the decaying burgers turbulence that one class of independent variables corresponds exactly to the so called one step replica symmetry breaking solution in the replica language we argue that this universality class holds if the correlations are sufficiently weak and propose a conjecture on the level of correlations which leads to different universality classes

synchronization landscapes in small world connected computer networks motivated by a synchronization problem in distributed computing we studied a simple growth model on regular and small world networks embedded in one and two dimensions we find that the synchronization landscape corresponding to the progress of the individual processors exhibits kardar parisi zhang like kinetic roughening on regular networks with short range communication links although the processors on average progress at a nonzero rate their spread the width of the synchronization landscape diverges with the number of nodes desynchronized state hindering efficient data management when random communication links are added on top of the one and two dimensional regular networks resulting in a small world network large fluctuations in the synchronization landscape are suppressed and the width approaches a finite value in the large system size limit synchronized state in the resulting synchronization scheme the processors make close to uniform progress with a nonzero rate without global intervention we obtain our results by simulating the simulations based on the exact algorithmic rules supported by coarse grained arguments

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

effect of changing data size on eigenvalues in the korean and japanese stock markets abstract in this study we attempted to determine how eigenvalues change according to random matrix theory rmt in stock market data as the number of stocks comprising the correlation matrix changes specifically we tested for changes in the eigenvalue properties as a function of the number and type of stocks in the correlation matrix we determined that the value of the eigenvalue increases in proportion with the number of stocks furthermore we noted that the largest eigenvalue maintains its identical properties regardless of the number and type whereas other eigenvalues evidence different features

a new approach to quantify power law cross correlation and its application to commodity markets abstract we proposed a new method detrended moving average cross correlation analysis dmca to detect the power law cross correlation between two correlated non stationary time series by combining detrended cross correlation analysis dcca and detrended moving average dma in order to compare the performance of dmca and dcca in the detection of cross correlation and to estimate the influence of periodic trend we generate two cross correlated time series x i and y i by a periodic two component fractionally autoregressive integrated moving average arfima process then we apply both methods to quantify the cross correlations of the generated series whose theoretical values are already known to us by comparing the results we obtained we find that the performance of this new approach is comparable to dcca with less calculating amounts our method can also reduce the impact of trends furthermore dmca for background and forward moving average case outperforms dcca in more accurate estimation when the analyzed times series are short in length to provide an example we also apply this new method to the time series of the real world data from brent and wti crude oil spot markets to investigate the complex cross market correlation between these commodity markets in all our method is another practical choice to detect the cross correlation between two short period non stationary time series and has potential application to real world problems

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

probing the tails of the ground state energy distribution for the directed polymer in a random medium of dimension d 1 2 3 via a monte carlo procedure in the disorder in order to probe with high precision the tails of the ground state energy distribution of disordered spin systems k orner katzgraber and hartmann have recently proposed an importance sampling monte carlo markov chain in the disorder in this paper we combine their monte carlo procedure in the disorder with exact transfer matrix calculations in each sample to measure the negative tail of ground state energy distribution p d e 0 for the directed polymer in a random medium of dimension d 1 2 3 in d 1 we check the validity of the algorithm by a direct comparison with the exact result namely the tracy widom distribution in dimensions d 2 and d 3 we measure the negative tail up to ten standard deviations which correspond to probabilities of order p d e 0 ensuremath sim 10 ensuremath 22 our results are in agreement with zhang s argument stating that the negative tail exponent ensuremath eta d of the asymptotic behavior mathrm ln phantom rule 0 2em 0ex p d e 0 ensuremath sim ensuremath ensuremath mid e 0 ensuremath mid ensuremath eta d as e 0 ensuremath rightarrow ensuremath ensuremath infty is directly related to the fluctuation exponent ensuremath theta d which governs the fluctuations ensuremath delta e 0 l ensuremath sim l ensuremath theta d of the ground state energy e 0 for polymers of length l via the simple formula ensuremath eta d 1 1 ensuremath ensuremath theta d throughout the paper we comment on the similarities and differences with spin glasses

economic fluctuations and statistical physics quantifying extremely rare and less rare events in finance one challenge of economics is that the systems treated by these sciences have no perfect metronome in time and no perfect spatial architecture crystalline or otherwise nonetheless as if by magic out of nothing but randomness one finds remarkably fine tuned processes in time we present an overview of recent research joining practitioners of economic theory and statistical physics to try to better understand puzzles regarding economic fluctuations one of these puzzles is how to describe outliers phenomena that lie outside of patterns of statistical regularity we review evidence consistent with the possibility that such outliers may not exist this possibility is supported by recent analysis of databases containing information about each trade of every stock r 2007 elsevier b v all rights reserved

statistics of extremal intensities for gaussian interfaces the extremal fourier intensities are studied for stationary edwards wilkinson type gaussian interfaces with power law dispersion we calculate the probability distribution of the maximal intensity and find that generically it does not coincide with the distribution of the integrated power spectrum i e roughness of the surface nor does it obey any of the known extreme statistics limit distributions the fisher tippett gumbel limit distribution is however recovered in three cases i in the nondispersive white noise limit ii for high dimensions and iii when only short wavelength modes are kept in the last two cases the limit distribution emerges in nonconventional scenarios

the square lattice f model revisited a loop cluster update scaling study the six vertex f model on the square lattice constitutes the unique example of an exactly solved model exhibiting an infinite order phase transition of the kosterlitz thouless type as one of the few non trivial exactly solved models it provides a welcome gauge for new numerical simulation methods and scaling techniques in view of the notorious problems of clearly resolving the kosterlitz thouless scenario in the two dimensional xy model numerically the f model in particular constitutes an instructive reference case for the simulational description of this type of phase transition we present a loop cluster update monte carlo study of the square lattice f model with a focus on the properties not exactly known such as the polarizability or the scaling dimension in the critical phase for the analysis of the simulation data finite size scaling is explicitly derived from the exact solution and plausible assumptions guided by the available exact results the careful inclusion of correction terms in the scaling formulae allows for a reliable determination of the asymptotic behaviour

a topos for algebraic quantum theory the aim of this paper is to relate algebraic quantum mechanics to topos theory so as to construct new foundations for quantum logic and quantum spaces moti vated by bohr s idea that the empirical content of quantum physics is accessible only through classical physics we show how a noncommutative c algebra of observables a induces a topos t a in which the amalgamation of all of its commutative subalgebras comprises a single commutative c algebra a according to the constructive gelfand duality theorem of banaschewski and mulvey the latter has an internal spectrum a in t a which in our approach plays the role of the quantum phase space of the sys tem thus we associate a locale which is the topos theoretical notion of a space and which intrinsically carries the intuitionistic logical structure of a heyting algebra to a c algebra which is the noncommutative notion of a space in this setting states on a become probability measures more precisely valuations on and self adjoint elements of a define continuous functions more precisely locale maps fromto scott s interval domain noting that open subsets of a correspond to propositions about the system the pairing map that assigns a generalized truth value to a state and a proposition assumes an extremely simple categorical form formulated in this way the quantum theory defined by a is essentially turned into a classical theory internal to the topos t a these results were inspired by the topos theoretic approach to quantum physics pro posed by butterfield and isham as recently generalized by doring and isham

universal large deviation function of the kardar parisi zhang equation in one dimension using the bethe ansatz we calculate the whole large deviation function of the displacement of particles in the asymmetric simple exclusion process asep on a ring when the size of the ring is large the central part of this large deviation function takes a scaling form independent of the density of particles we suggest that this scaling function found for the asep is universal and should be characteristic of all the systems described by the kardar parisi zhang equation in 1 1 dimension simulations done on two simple growth models are in reasonable agreement with this conjecture

visibility graph analysis on quarterly macroeconomic series of china based on complex network theory abstract the visibility graph approach and complex network theory provide a new insight into time series analysis the inheritance of the visibility graph from the original time series was further explored in the paper we found that degree distributions of visibility graphs extracted from pseudo brownian motion series obtained by the frequency domain algorithm exhibit exponential behaviors in which the exponential exponent is a binomial function of the hurst index inherited in the time series our simulations presented that the quantitative relations between the hurst indexes and the exponents of degree distribution function are different for different series and the visibility graph inherits some important features of the original time series further we convert some quarterly macroeconomic series including the growth rates of value added of three industry series and the growth rates of gross domestic product series of china to graphs by the visibility algorithm and explore the topological properties of graphs associated from the four macroeconomic series namely the degree distribution and correlations the clustering coefficient the average path length and community structure based on complex network analysis we find degree distributions of associated networks from the growth rates of value added of three industry series are almost exponential and the degree distributions of associated networks from the growth rates of gdp series are scale free we also discussed the assortativity and disassortativity of the four associated networks as they are related to the evolutionary process of the original macroeconomic series all the constructed networks have small world features the community structures of associated networks suggest dynamic changes of the original macroeconomic series we also detected the relationship among government policy changes community structures of associated networks and macroeconomic dynamics we find great influences of government policies in china on the changes of dynamics of gdp and the three industries adjustment the work in our paper provides a new way to understand the dynamics of economic development

width distribution of curvature driven interfaces a study of universality one dimensional interfaces with curvature driven growth kinetics are investigated we calculate the steady state distribution p mathit w 2 of the square of the width of the interface mathit w 2 and show that as in the case for random walk interfaces the result can be written in a scaling form mathit w 2 p mathit w 2 ensuremath phi mathit w 2 mathit w 2 where mathit w 2 is the average of mathit w 2 the scaling function ensuremath phi x is found to be distinct from that of random walk interfaces but as our monte carlo simulations indicate this function is universal for curvature driven growth it is argued that comparison of scaling functions can be a useful method for distinguishing between universality classes of growth processes

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

football goal distributions and extremal statistics we analyse the distributions of the number of goals scored by home teams away teams and the total scored in the match in domestic football games from 169 countries between 1999 and 2001 the probability density functions pdfs of goals scored are too heavy tailed to be fitted over their entire ranges by poisson or negative binomial distributions which would be expected for uncorrelated processes log normal distributions cannot include zero scores and here we find that the pdfs are consistent with those arising from extremal statistics in addition we show that it is sufficient to model english top division and fa cup matches in the seasons of 1970 71 2000 01 on poisson or negative binomial distributions as reported in analyses of earlier seasons and that these are not consistent with extremal statistics

random matrix approach to categorical data analysis correlation and similarity measures are widely used in all the areas of sciences and social sciences often the variables are not numbers but are instead qualitative descriptors called categorical data we define and study similarity matrix as a measure of similarity for the case of categorical data this is of interest due to a deluge of categorical data such as movie ratings top 10 rankings and data from social media in the public domain that require analysis we show that the statistical properties of the spectra of similarity matrices constructed from categorical data follow random matrix predictions with the dominant eigenvalue being an exception we demonstrate this approach by applying it to the data for indian general elections and sea level pressures in the north atlantic ocean

localized short range correlations in the spectrum of the equal time correlation matrix we suggest a procedure to identify those parts of the spectrum of the equal time correlation matrix mathbf c where relevant information about correlations of a multivariate time series is induced using an ensemble average over each of the distances between eigenvalues all nearest neighbor distributions can be calculated individually we present numerical examples where a information about cross correlations is found in the so called bulk of eigenvalues which generally is thought to contain only random correlations and where b the information extracted from the lower edge of the spectrum of mathbf c is statistically more significant than that extracted from the upper edge we apply the analysis to electroencephalographic recordings with epileptic events

the tate conjecture for k3 surfaces in odd characteristic we show that the classical kuga satake construction gives rise away from characteristic 2 to an open immersion from the moduli of primitively polarized k3 surfaces of any fixed degree to a certain regular integral model for a shimura variety of orthogonal type this allows us to attach to every polarized k3 surface in odd characteristic an abelian variety such that divisors on the surface can be identified with certain endomorphisms of the attached abelian variety in turn this reduces the tate conjecture for k3 surfaces over finitely generated fields of odd characteristic to a version of the tate conjecture for certain endomorphisms on the attached kuga satake abelian variety which we prove as a by product of our methods we also show that the moduli stack of primitively polarized k3 surfaces of degree 2d is quasi projective and when d is not divisible by p 2 is geometrically irreducible in characteristic p we indicate how the same method applies to prove the tate conjecture for co dimension 2 cycles on cubic fourfolds

scanning the critical fluctuations application to the phenomenology of the two dimensional xy model we show how applying field conjugated to the order parameter may act as a very precise probe to explore the probability distribution function of the order parameter using this magnetic field scanning on large scale numerical simulations of the critical two dimensional xy model we are able to discard the conjectured double exponential form of the large magnetization asymptote

correlation and network analysis of global financial indices random matrix theory rmt and network methods are applied to investigate the correlation and network properties of 20 financial indices the results are compared before and during the financial crisis of 2008 in the rmt method the components of eigenvectors corresponding to the second largest eigenvalue form two clusters of indices in the positive and negative directions the components of these two clusters switch in opposite directions during the crisis the network analysis uses the fruchterman reingold layout to find clusters in the network of indices at different thresholds at a threshold of 0 6 before the crisis financial indices corresponding to the americas europe and asia pacific form separate clusters on the other hand during the crisis at the same threshold the american and european indices combine together to form a strongly linked cluster while the asia pacific indices form a separate weakly linked cluster if the value of the threshold is further increased to 0 9 then the european indices france germany and the united kingdom are found to be the most tightly linked indices the structure of the minimum spanning tree of financial indices is more starlike before the crisis and it changes to become more chainlike during the crisis the average linkage hierarchical clustering algorithm is used to find a clearer cluster structure in the network of financial indices the cophenetic correlation coefficients are calculated and found to increase significantly which indicates that the hierarchy increases during the financial crisis these results show that there is substantial change in the structure of the organization of financial indices during a financial crisis

universal fluctuations of the aex index abstract we compute the analytic expression of the probability distributions f a e x and f a e x of the normalized positive and negative aex netherlands index daily returns r t furthermore we define the re scaled aex daily index positive returns r t and negative returns r t which we call after normalization the positive fluctuations and negative fluctuations we use the kolmogorov smirnov statistical test as a method to find the values of that optimize the data collapse of the histogram of the fluctuations with the bramwell holdsworth pinton bhp probability density function the optimal parameters that we found are 0 46 and 0 43 since the bhp probability density function appears in several other dissimilar phenomena our result reveals a universal feature of stock exchange markets

toposes as homotopy groupoids the purpose of this paper is to deepen our understanding of toposes as generalized spaces we prove that for any grothendieck topos 8 there exists a space x and a collection p of paths in x so that 6 is equivalent to the category of sheaves on x which are constant along the paths in p furthermore by taking homotopy classes of paths in p we obtain a continuous groupoid g 3 x so that the sheaves on x which are acted upon by g are exactly those sheaves which are constant along the paths in p as a consequence we thus deduce one of the main results proved in jtj showing that any topos can be described as the category of equivariant sheaves for some continuous groupoid our construction has the following additional properties the covering map cp x 6 is connected and locally connected and tiberwise contractible this implies that cp induces a full embedding of a into the category of sheaves on a which preserves exponentials in addition to tirst order logic moreover it implies that cp induces an isomorphism between the etale cohomology groups of the topos 8 and the cohomology of the space x this last aspect will be discussed in a different paper

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

time scales involved in emergent market coherence abstract in addressing the question of the time scales characteristic for the market formation we analyze high frequency tick by tick data from the nyse and from the german market by using returns on various time scales ranging from seconds or minutes up to 2 days we compare magnitude of the largest eigenvalue of the correlation matrix for the same set of securities but for different time scales for various sets of stocks of different capitalization and the average trading frequency we observe a significant elevation of the largest eigenvalue with increasing time scale our results from the correlation matrix study can be considered as a manifestation of the so called epps effect there is no unique explanation of this effect and it seems that many different factors play a role here one of such factors is randomness in transaction moments for different stocks another interesting conclusion to be drawn from our results is that in the contemporary markets the emergence of significant correlations occurs on time scales much smaller than in the more distant history

scaling and commonality in anomalous fluctuation statistics in models for turbulence and ferromagnetism recently portelli et al 2003 phys rev lett 90 104501 have semi numerically obtained a functional form of the probability distribution of fluctuations in the total energy flow in a model for fluid turbulence this follows earlier work suggesting that fluctuations in the total magnetization in the 2d xy model for a ferromagnet also follow this distribution here starting from the scaling ansatz that is the basis of the turbulence model we analytically derive the functional form of this distribution and find its single control parameter that depends upon the scaling exponents and system size of the model our analysis allows us to identify this explicitly with that of the xy model and suggest a possible generalization

numerical study of the two replica overlap of the 3d edwards anderson ising spin glass abstract we present results of recent high statistics monte carlo simulations of the edwards anderson ising spin glass model in three dimensions the study is based on a non boltzmann sampling technique the multi self overlap algorithm which is specifically tailored for sampling rare event states we thus concentrate on those properties which are difficult to obtain with standard canonical boltzmann sampling such as the free energy barriers f b q in the probability densities p j q of the parisi overlap parameter q and the behavior of the tails of the disorder averaged density p q p j q av our results for the tails disagree with mean field predictions and support extreme order statistics over many orders of magnitude

detection and characterization of changes of the correlation structure in multivariate time series we propose a method based on the equal time correlation matrix as a sensitive detector for phase shape correlations in multivariate data sets the key point of the method is that changes of the degree of synchronization between time series provoke level repulsions between eigenstates at both edges of the spectrum of the correlation matrix consequently detailed information about the correlation structure of the multivariate data set is imprinted into the dynamics of the eigenvalues and into the structure of the corresponding eigenvectors the performance of the technique is demonstrated by application to n f tori autoregressive models and coupled chaotic systems the high sensitivity the comparatively small computational effort and the excellent time resolution of the method recommend it for application to the analysis of complex spatially extended nonstationary systems

universal features of the order parameter fluctuations reversible and irreversible aggregation we discuss the universal scaling laws of order parameter fluctuations in any system in which a second order critical behavior can be identified these scaling laws can be derived rigorously for equilibrium systems when combined with a finite size scaling analysis the relation between the order parameter the criticality and the scaling law of fluctuations has been established and the connection between the scaling function and the critical exponents has been found we give examples in out of equilibrium aggregation models such as the smoluchowski kinetic equations or at equilibrium ising and percolation models

general limit distributions for sums of random variables with a matrix product representation the general limit distributions of the sum of random variables described by a finite matrix product ansatz are characterized using a mapping to a hidden markov chain formalism non standard limit distributions are obtained and related to a form of ergodicity breaking in the underlying non homogeneous hidden markov chain the link between ergodicity and limit distributions is detailed and used to provide a full algorithmic characterization of the general limit distributions

statistical self similarity of one dimensional growth processes abstract for one dimensional growth processes we consider the distribution of the height above a given point of the substrate and study its scale invariance in the limit of large times we argue that for self similar growth from a single seed the universal distribution is the tracy widom distribution from the theory of random matrices and that for the growth from a flat substrate it is some other only numerically determined distribution in particular for the polynuclear growth model in the droplet geometry the height maps onto the longest increasing subsequence of a random permutation from which the height distribution is identified as the tracy widom distribution

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

multifractal detrended cross correlation analysis of gold price and sensex abstract this paper studies the variation of autocorrelation and cross correlation coefficients of gold price and sensex fluctuations with time the paper uses mfdfa and mfdxa methodologies sensex plays a more dominant role in the variation of cross correlations it is observed that the cross correlation coefficients can be linked with the stability of the market the market is most stable when the two series are most correlated

collective behavior of stock price movements a random matrix theory approach we review recent work on quantifying collective behavior among stocks by applying the conceptual framework of random matrix theory rmt developed in physics to describe the energy levels ofcomplex systems rmt makes predictions f universal properties that do not depend on the interactions between the elements comprising the system deviations from rmt provide clues regarding system speci c properties we compare the statistics ofthe cross correlation matrix c whose elements cij are the correlation coe cients ofprice uctuations ofstock i and j against a random matrix having the same symmetry properties it is found that rmt methods can distinguish random and non random parts of c the non random part of c which deviates from rmt results provides information regarding genuine collective behavior among stocks c 2001 published by elsevier science b v

dynamics of market correlations taxonomy and portfolio analysis the time dependence of the recently introduced minimum spanning tree description of correlations between stocks called the asset tree has been studied in order to reflect the financial market taxonomy the nodes of the tree are identified with stocks and the distance between them is a unique function of the corresponding element of the correlation matrix by using the concept of a central vertex chosen as the most strongly connected node of the tree an important characteristic is defined by the mean occupation layer during crashes due to the strong global correlation in the market the tree shrinks topologically and this is shown by a low value of the mean occupation layer the tree seems to have a scale free structure where the scaling exponent of the degree distribution is different for business as usual and crash periods the basic structure of the tree topology is very robust with respect to time we also point out that the diversification aspect of portfolio optimization results in the fact that the assets of the classic markowitz portfolio are always located on the outer leaves of the tree technical aspects such as the window size dependence of the investigated quantities are also discussed

quantifying economic fluctuations abstract this manuscript is a brief summary of a talk designed to address the question of whether two of the pillars of the field of phase transitions and critical phenomena scale invariance and universality can be useful in guiding research on interpreting empirical data on economic fluctuations using this conceptual framework as a guide we empirically quantify the relation between trading activity measured by the number of transactions n and the price change g t for a given stock over a time interval t t t we relate the time dependent standard deviation of price changes volatility to two microscopic quantities the number of transactions n t in t and the variance w2 t of the price changes for all transactions in t we find that the long ranged volatility correlations are largely due to those of n we then argue that the tail exponent of the distribution of n is insufficient to account for the tail exponent of p g x since n and w display only weak inter dependency our results show that the fat tails of the distribution p g x arises from w finally we review recent work on quantifying collective behavior among stocks by applying the conceptual framework of random matrix theory rmt rmt makes predictions for universal properties that do not depend on the interactions between the elements comprising the system and deviations from rmt provide clues regarding system specific properties we compare the statistics of the cross correlation matrix c whose elements cij are the correlation coefficients of price fluctuations of stock i and j against a random matrix having the same symmetry properties it is found that rmt methods can distinguish random and non random parts of c the non random part of c which deviates from rmt results provides information regarding genuine collective behavior among stocks we also discuss results that are reminiscent of phase transitions in spin systems where the divergent behavior of the response function at the critical point zero magnetic field leads to large fluctuations and we discuss a curious symmetry breaking a feature qualitatively identical to the behavior of the probability density of the magnetization for fixed values of the inverse temperature

anomalous scaling for passively advected magnetic fields the kraichnan model 1 for passive scalar advection is by now considered a paradigm for intermittency problems a first systematic understanding of the scalar field scaling properties has been attained the crucial property of the model is that equal time correlation functions obey closed equations of motion this stems from the small correlation time of the velocity field advecting the scalar it has been recognized that a general mechanism for intermittency and anomalous scaling is the presence of nontrivial zero modes of the closed equations satisfied by the correlation functions 2 3 4 such mechanism can be illustrated in its simplest form by considering the following linear differential equation frac d 2 y r d r 2 frac a r frac dy r dr b frac y r r 2 f frac r l 1 where r varies on the positive axis y r is the unknown function a and b are two constants and f r l varies over distances of order l and is almost constant for r l in the physical situations l is the integral scale and the mechanisms maintaining the turbulence are supposed to act at this scale as for intermittency in navier stokes turbulence we are interested in the scaling behavior in the inertial range i e for r l note that the l h s of 1 is invariant under scale transformations reflecting the scale invariance of the general solution of 1 is given by a non homogeneous and a linear combination of two homogeneous solutions the combination is fixed by the conditions that y r being a correlation function should be regular and vanish at infinity

power fluctuations in a closed turbulent shear flow we study experimentally the power consumption p of a confined turbulent flow at constant reynolds number re we analyze in details its temporal dynamics and statistical properties in a setup that covers two decades in reynolds numbers we show that nontrivial power fluctuations occur over a wide range of amplitudes and that they involve coherent fluid motions over the entire system size as a result the power fluctuations do not result from averaging of independent subsystems and its probability density function pi p is strongly non gaussian the shape of pi p is reynolds number independant and we show that the relative intensity of fluctuations decreases very slowly as re increases these results are discussed in terms of an analogy with critical phenomena

global fluctuations and gumbel statistics we explain how the statistics of global observables in correlated systems can be related to extreme value problems and to gumbel statistics this relationship then naturally leads to the emergence of the generalized gumbel distribution ga x with a real index a in the study of global fluctuations to illustrate these findings we introduce an exactly solvable nonequilibrium model describing an energy flux on a lattice with local dissipation in which the fluctuations of the global energy are precisely described by the generalized gumbel distribution

similarities between stock price correlation networks and co main product networks threshold scenarios abstract because of the high yields and high risks associated with the stock market investors can hold diversified portfolios with low relativity of stocks to reduce unsystematic risk the current literature analyzes single factors affecting the relativity of stocks but in this paper we analyze the correlations between different factors to provide multiple perspectives of and about investment portfolios this study analyzes the relationships between the similarities of the main products of listed companies and the varying degrees of correlations of stock price by examining different threshold scenarios of the energy industry between 2012 and 2016 and then constructing stock price correlation threshold networks and co main product networks to analyze the similarities in their structures the results indicate that two factors are significantly correlated in 97 5 of the scenarios and that these factors are the most strongly correlated when the threshold is between 0 5 and 0 7 the two networks exhibit a high degree of similarity in degree weighted degree and community division main product similarity used as supplementary information for stock relativity research plays a role similar to stock price correlations in certain scenarios furthermore compared with stock price correlations the similarity of main products is simpler and more intuitive this paper proposes a new method to study stock relativity based on different threshold scenarios thus it could serve as a reference for investors when developing portfolio strategies from multiple perspectives

an analysis of cross correlations in an emerging market abstract we apply random matrix theory to compare correlation matrix estimators c obtained from emerging market data the correlation matrices are constructed from 10 years of daily data for stocks listed on the johannesburg stock exchange jse from january 1993 to december 2002 we test the spectral properties of c against random matrix predictions and find some agreement between the distributions of eigenvalues nearest neighbour spacings distributions of eigenvector components and the inverse participation ratios for eigenvectors we show that interpolating both missing data and illiquid trading days with a zero order hold increases agreement with rmt predictions for the more realistic estimation of correlations in an emerging market we suggest a pairwise measured data correlation matrix for the data set used this approach suggests greater temporal stability for the leading eigenvectors an interpretation of eigenvectors in terms of trading strategies is given as opposed to classification by economic sectors

the surface statistics of a granular aggregate the problem of the surface fluctuations in a settled granular material is posed a simple model is given which describes the process by which a particle settles and comes to rest on the existing surface of the packing and from this a set of langevin equations for the fourier modes of the surface are derived these equations imply that the fourier amplitudes behave like the velocities of a set of independent brownian particles we show that this results in logarithmically divergent surface fluctuations if the flux of particles onto the surface is random the divergence being removed by a more accurate description of the settling material for example by having the granules fall through a sieve

record statistics for multiple random walks we study the statistics of the number of records r n n for n identical and independent symmetric discrete time random walks of n steps in one dimension all starting at the origin at step 0 at each time step each walker jumps by a random length drawn independently from a symmetric and continuous distribution we consider two cases i when the variance ensuremath sigma 2 of the jump distribution is finite and ii when ensuremath sigma 2 is divergent as in the case of l evy flights with index 0l ensuremath mu l2 in both cases we find that the mean record number ensuremath langle r n n ensuremath rangle grows universally as ensuremath sim ensuremath alpha n sqrt n for large n but with a very different behavior of the amplitude ensuremath alpha n for ng1 in the two cases we find that for large n ensuremath alpha n ensuremath approx 2 sqrt mathrm ln n independently of ensuremath sigma 2 in case i in contrast in case ii the amplitude approaches to an n independent constant for large n ensuremath alpha n ensuremath approx 4 sqrt ensuremath pi independently of 0l ensuremath mu l2 for finite ensuremath sigma 2 we argue and this is confirmed by our numerical simulations that the full distribution of r n n sqrt n ensuremath 2 sqrt mathrm ln n sqrt mathrm ln n converges to a gumbel law as n ensuremath rightarrow ensuremath infty and n ensuremath rightarrow ensuremath infty in case ii our numerical simulations indicate that the distribution of r n n sqrt n converges for n ensuremath rightarrow ensuremath infty and n ensuremath rightarrow ensuremath infty to a universal nontrivial distribution independently of ensuremath mu we discuss the applications of our results to the study of the record statistics of 366 daily stock prices from the standard poor s 500 index

universal magnetic fluctuations with a field induced length scale we calculate the probability density function for the order parameter fluctuations in the low temperature phase of the two dimensional mathrm xy model of magnetism near the line of critical points a finite correlation length ensuremath xi is introduced with a small magnetic field h and an expression for ensuremath xi h is developed by treating nonlinear contributions to the field energy using a hartree approximation we find analytically a series of universal non gaussian distributions of the finite size scaling form p m l ensuremath xi ensuremath sim l ensuremath beta ensuremath nu p l ml ensuremath beta ensuremath nu ensuremath xi l and present a function of the form p x ensuremath sim mathrm exp x ensuremath mathrm exp x a h that gives the probability density functions to an excellent approximation we propose a h as an indirect measure of the length scale of correlations in a wide range of complex systems

network trending leadership followership and neutrality among companies a random matrix approach abstract in this article we analyze the cross correlation between returns of different stocks to answer the following important questions the first one is if there exists collective behavior in a financial market how could we detect it and the second question is is there a particular company among the companies of a market as the leader of the collective behavior or is there no specified leadership governing the system similar to some complex systems we use the method of random matrix theory to answer the mentioned questions cross correlation matrix of index returns of four different markets is analyzed the participation ratio quantity related to each matrices eigenvectors and the eigenvalue spectrum is calculated we introduce shuffled matrix created of cross correlation matrix in such a way that the elements of the later one are displaced randomly comparing the participation ratio quantities obtained from a correlation matrix of a market and its related shuffled one on the bulk distribution region of the eigenvalues we detect a meaningful deviation between the mentioned quantities indicating the collective behavior of the companies forming the market by calculating the relative deviation of participation ratios we obtain a measure to compare the markets according to their collective behavior answering the second question we show there are three groups of companies the first group having higher impact on the market trend called leaders the second group is followers and the third one is the companies who have not a considerable role in the trend the results can be utilized in portfolio construction