spectra of complex networks we propose a general approach to the description of spectra of complex networks for the spectra of networks with uncorrelated vertices and a local treelike structure exact equations are derived these equations are generalized to the case of networks with correlations between neighboring vertices the tail of the density of eigenvalues ensuremath rho ensuremath lambda at large ensuremath lambda is related to the behavior of the vertex degree distribution p k at large k in particular as p k ensuremath sim k ensuremath ensuremath gamma ensuremath rho ensuremath lambda ensuremath sim ensuremath lambda 1 ensuremath 2 ensuremath gamma we propose a simple approximation which enables us to calculate spectra of various graphs analytically we analyze spectra of various complex networks and discuss the role of vertices of low degree we show that spectra of locally treelike random graphs may serve as a starting point in the analysis of spectral properties of real world networks e g of the internet

levy ornstein uhlenbeck and subordination spectral vs jump description unlike brownian motion which propagates diffusively and whose sample path trajectories are continuous non brownian levy motions propagate via jumps flights and their sample path trajectories are purely discontinuous when analyzing systems involving non brownian levy motions the common practice is to use either spectral or fractional calculus methods in this manuscript we suggest an alternative analytical approach using the poisson superposition jump structure of non brownian levy motions we demonstrate this approach in two exemplary topics i systems governed by l evy driven ornstein uhlenbeck dynamics and ii systems subject to temporal levy subordination we show that this approach yields answers and insights that are not attainable using spectral methods alone

power law shot noise the behavior of power law shot noise for which the associated impulse response functions assume a decaying power law form is explored expressions are obtained for the moments moment generating functions amplitude probability density functions autocorrelation functions and power spectral densities for a variety of parameters of the process for certain parameters the power spectral density exhibits 1 f type behavior over a substantial range of frequencies so that the process serves as a source of 1 f sup alpha shot noise for alpha in the range 0

critical percolation and the minimal spanning tree in slabs the minimal spanning forest on d is known to consist of a single tree for d 2 and is conjectured to consist of infinitely many trees for large d in this paper we prove that there is a single tree for quasi planar graphs such as 2 0 k d 2 our method relies on generalizations of the gluing lemma of duminil copin sidoravicius and tassion a related result is that critical bernoulli percolation on a slab satisfies the box crossing property its proof is based on a new russo seymour welsh type theorem for quasi planar graphs thus at criticality the probability of an open path from 0 of diameter n decays polynomially in n this strengthens the result of duminil copin et al where the absence of an infinite cluster at criticality was first established 2017 wiley periodicals inc

connection probabilities and rsw type bounds for the two dimensional fk ising model we prove russo seymour welsh type uniform bounds on crossing probabilities for the fk ising fk percolation with cluster weight q 2 model at criticality independent of the boundary conditions our proof relies mainly on smirnov s fermionic observable for the fk ising model 24 which allows us to get precise estimates on boundary connection probabilities we stay in a discrete setting in particular we do not make use of any continuum limit and our result can be used to derive directly several noteworthy properties including some new ones among which are the fact that there is no infinite cluster at criticality tightness properties for the interfaces and the existence of several critical exponents in particular the half plane one arm exponent such crossing bounds are also instrumental for important applications such as constructing the scaling limit of the ising spin field 6 and deriving polynomial bounds for the mixing time of the glauber dynamics at criticality 17

random matrix analysis of complex networks we study complex networks under random matrix theory rmt framework using nearest neighbor and next nearest neighbor spacing distributions we analyze the eigenvalues of the adjacency matrix of various model networks namely random scale free and small world networks these distributions follow the gaussian orthogonal ensemble statistic of rmt to probe long range correlations in the eigenvalues we study spectral rigidity via the ensuremath delta 3 statistic of rmt as well it follows rmt prediction of linear behavior in semilogarithmic scale with the slope being ensuremath sim 1 ensuremath pi 2 random and scale free networks follow rmt prediction for very large scale a small world network follows it for sufficiently large scale but much less than the random and scale free networks

statistical resilience of random populations to random perturbations we consider populations represented by random collections of real valued points and explore their statistical resilience to random perturbations seeking populations whose statistics remain qualitatively unchanged by the action of arbitrary random perturbations of a certain type studying a general physical perturbation scheme we obtain an explicit characterization of statistically resilient populations show that these objects are fractal and comprehensively analyze their topological and statistical structures an application of statistical resilience attained is an alternative explanation of the ubiquity of power law statistics

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

describing the indescribable the stationary structures of transient markovian dynamics markovian dynamics are the most elemental and prevalent form of stochastic dynamics in science and engineering the statistical evolution of given markovian dynamics is governed by its master equation and the integrability of the master equation s positive valued stationary solutions determines the dynamics statistical behavior integrability implies that the dynamics converge to steady state and non integrability implies that the dynamics are transient this letter establishes that all the positive valued stationary solutions of the master equation represent poissonian steady states poissonian steady states generalize the notion of regular steady states and quantify the stationary structures of general markovian dynamics in particular we unveil and quantify the stationary structures of transient markovian dynamics

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

discontinuity of the magnetization in one dimensional 1 x y 2 ising and potts models results from percolation theory are used to study phase transitions in one dimensional ising and q state potts models with couplings of the asymptotic form j sub x y approx const absolute value x y sup 2 for translation invariant systems with well defined lim sub x infinity x sup 2 j sub x j sup possibly o or infinity we establish 1 there is no long range order at inverse temperatures beta with beta j sup less than or equal to1 2 if beta j sup q then by sufficiently increasing j sub 1 the spontaneous magnetization m is made positive 3 in models with 0 j sup infinity the magnetization is discontinuous at the transition point as originally predicted by thouless and obeys m beta sub c less than or equal to 1 beta sub c j sup sup 1 2 4 for ising q 2 models with j sup infinity it is noted that the correlation function decays as beta approx c beta absolute value x y sup 2 whenever beta beta sub c points 1 3 are deduced from previous percolation results by utilizing the fortuin kasteleyn representation which also yields other results of independent interest relating potts models with different values of q

universality of accelerating change abstract on large time scales the progress of human technology follows an exponential growth trend that is termed accelerating change the exponential growth trend is commonly considered to be the amalgamated effect of consecutive technology revolutions where the progress carried in by each technology revolution follows an s curve and where the aging of each technology revolution drives humanity to push for the next technology revolution thus as a collective mankind is the intelligent designer of accelerating change in this paper we establish that the exponential growth trend and only this trend emerges universally on large time scales from systems that combine together two elements randomness and amalgamation hence the universal generation of accelerating change can be attained by systems with no intelligent designer

gini characterization of extreme value statistics abstract this paper presents a profound connection between gini s index and extreme value statistics gini s index is a quantitative gauge for the evenness of probability laws defined on the positive half line and is the common measure of societal egalitarianism applied in economics and in the social sciences extreme value statistics namely the gumbel frechet and weibull probability laws are the only possible asymptotic statistics emerging from the extremes of large ensembles of independent and identically distributed random variables extreme value statistics play a major role all across science and engineering in the analysis of rare and extreme events introducing generalizations of gini s index and exploring an elemental poissonian structure underlying the extreme value statistics we establish in this paper a gini based characterization of extreme value statistics

random matrix approach to cross correlations in financial data we analyze cross correlations between price fluctuations of different stocks using methods of random matrix theory rmt using two large databases we calculate cross correlation matrices c of returns constructed from i 30 min returns of 1000 us stocks for the 2 yr period 1994 1995 ii 30 min returns of 881 us stocks for the 2 yr period 1996 1997 and iii 1 day returns of 422 us stocks for the 35 yr period 1962 1996 we test the statistics of the eigenvalues lambda i of c against a null hypothesis a random correlation matrix constructed from mutually uncorrelated time series we find that a majority of the eigenvalues of c fall within the rmt bounds lambda lambda for the eigenvalues of random correlation matrices we test the eigenvalues of c within the rmt bound for universal properties of random matrices and find good agreement with the results for the gaussian orthogonal ensemble of random matrices implying a large degree of randomness in the measured cross correlation coefficients further we find that the distribution of eigenvector components for the eigenvectors corresponding to the eigenvalues outside the rmt bound display systematic deviations from the rmt prediction in addition we find that these deviating eigenvectors are stable in time we analyze the components of the deviating eigenvectors and find that the largest eigenvalue corresponds to an influence common to all stocks our analysis of the remaining deviating eigenvectors shows distinct groups whose identities correspond to conventionally identified business sectors finally we discuss applications to the construction of portfolios of stocks that have a stable ratio of risk to return less

on 1 f noise and other distributions with long tails abstract it is shown following shockley shockley w 1957 proc ire 45 279 290 that when a population is engaged in tasks whose completion requires the successful conclusion of many independent subtasks the distribution function for successes in the primary task is log normal it is also shown that when the dispersion of the log normal distribution is large the distribution is mimicked by a 1 x distribution over a wide range of x this argument provides a generic set of processes that yields the much observed 1 x distribution and will also lead to a 1 f noise spectrum it is commonly found that distributions that seem to be log normal over a broad range say to the 95th percentile of a population change to an inverse fractional power pareto distribution for the last few percentile annual income distributions are examples with this structure the very wealthy generally achieve their superwealth through amplification processes that are not available to most we have introduced a simple amplification model to characterize the transition from a log normal distribution to an inverse power pareto tail

spectra of random graphs with given expected degrees in the study of the spectra of power law graphs there are basically two competing approaches one is to prove analogues of wigner s semicircle law whereas the other predicts that the eigenvalues follow a power law distribution although the semicircle law and the power law have nothing in common we will show that both approaches are essentially correct if one considers the appropriate matrices we will prove that under certain mild conditions the eigenvalues of the normalized laplacian of a random power law graph follow the semicircle law whereas the spectrum of the adjacency matrix of a power law graph obeys the power law our results are based on the analysis of random graphs with given expected degrees and their relations to several key invariants of interest are a number of new values for the exponent where phase transitions for eigenvalue distributions occur the spectrum distributions have direct implications to numerous graph algorithms such as for example randomized algorithms that involve rapidly mixing markov chains

power law distributions in empirical data power law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man made phenomena unfortunately the detection and characterization of power laws is complicated by the large fluctuations that occur in the tail of the distribution the part of the distribution representing large but rare events and by the difficulty of identifying the range over which power law behavior holds commonly used methods for analyzing power law data such as least squares fitting can produce substantially inaccurate estimates of parameters for power law distributions and even in cases where such methods return accurate answers they are still unsatisfactory because they give no indication of whether the data obey a power law at all here we present a principled statistical framework for discerning and quantifying power law behavior in empirical data our approach combines maximum likelihood fitting methods with goodness of fit tests based on the kolmogorov smirnov ks statistic and likelihood ratios we evaluate the effectiveness of the approach with tests on synthetic data and give critical comparisons to previous approaches we also apply the proposed methods to twenty four real world data sets from a range of different disciplines each of which has been conjectured to follow a power law distribution in some cases we find these conjectures to be consistent with the data while in others the power law is ruled out

benford s law a poisson perspective abstract benford s law is a counterintuitive statistical law asserting that the distribution of leading digits taken from a large ensemble of positive numerical values that range over many orders of scale is logarithmic rather than uniform as intuition suggests in this paper we explore benford s law from a poisson perspective considering ensembles of positive numerical values governed by poisson process statistics we show that this poisson setting naturally accommodates benford s law and i establish a poisson characterization and a poisson multidigit extension of benford s law ii study a system invariant leading digit distribution which generalizes benford s law and establish a poisson characterization and a poisson multidigit extension of this distribution iii explore the universal emergence of the system invariant leading digit distribution couple this universal emergence to the universal emergence of the weibull and frechet extreme value distributions and distinguish the special role of benford s law in this universal emergence iv study the continued fractions counterpart of the system invariant leading digit distribution and establish a poisson characterization of this distribution and v unveil the elemental connection between the system invariant leading digit distribution and its continued fractions counterpart this paper presents a panoramic poisson approach to benford s law to its system invariant generalization and to its continued fractions counterpart

absence of infinite cluster for critical bernoulli percolation on slabs we prove that for bernoulli percolation on a graph 2 0 k k 0 there is no infinite cluster at criticality almost surely the proof extends to finite range bernoulli percolation models on 2 that are invariant under 2 rotation and reflection 2016 wiley periodicals inc

eigenvalue spectra of random matrices for neural networks the dynamics of neural networks is influenced strongly by the spectrum of eigenvalues of the matrix describing their synaptic connectivity in large networks elements of the synaptic connectivity matrix can be chosen randomly from appropriate distributions making results from random matrix theory highly relevant unfortunately classic results on the eigenvalue spectra of random matrices do not apply to synaptic connectivity matrices because of the constraint that individual neurons are either excitatory or inhibitory therefore we compute eigenvalue spectra of large random matrices with excitatory and inhibitory columns drawn from distributions with different means and equal or different variances

shot noise displaying simultaneously the noah and joseph effects shot noise processes displaying the noah effect are of infinite variance hence the measurement of the correlation structure of such processes indicating whether the joseph effect is displayed as well cannot be conducted via their autocovariance functions which are undefined to circumvent this problem a poisson based analysis applicable to shot noise processes with infinite variance as well as to shot noise processes with divergent noise levels is developed a poissonian autocovariance function which characterizes the process distribution of general shot noise processes is introduced in particular this function governs and quantifies the processes stationary structure amplitudal behavior and correlation structure temporal behavior the poissonian methodology developed enables a precise quantitative analysis of shot noise processes displaying simultaneously the noah and joseph effects

a note on the level spacings distribution of the hamiltonians in the transition region between integrability and chaos it is well known that there is no strict universality of the spectral fluctuations of quantum hamiltonians whose classical counterparts undergo the transition from integrability to complete chaos the author discusses the level spacings distribution p s and explains why the semiclassical formulae of berry and robnik cannot be correct for small s there is no global universality of p s for nearly integrable systems but the approach to the integrability as the perturbation parameter in goes to zero can be universal this is reflected in the fact that the slope of dp ds at s 0 for small in is universally inversely proportional to in the author gives two models in terms of two dimensional random matrices one of them being based on maximum entropy considerations the author also points out the connection to the statistics of zeros of random functions and discusses the numerical evidence

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

spectra of real world graphs beyond the semicircle law results on the spectra of adjacency matrices corresponding to models of real world graphs we find that when the number of links grows as the number of nodes the spectral density of uncorrelated random matrices does not converge to the semicircle law furthermore the spectra of real world graphs have specific features depending on the details of the corresponding models in particular scale free graphs develop a trianglelike spectral density with a power law tail while small world graphs have a complex spectral density consisting of several sharp peaks these and further results indicate that the spectra of correlated graphs represent a practical tool for graph classification and can provide useful insight into the relevant structural properties of real networks

maximum entropy formalism fractals scaling phenomena and 1 f noise a tale of tails in this report on examples of distribution functions with long tails we a show that the derivation of distributions with inverse power tails from a maximum entropy formalism would be a consequence only of an unconventional auxilliary condition that involves the specification of the average value of a complicated logarithmic function b review several models that yield log normal distributions c show that log normal distributions may mimic 1 f noise over a certain range and d present an amplification model to show how log normal personal income distributions are transformed into inverse power pareto distributions in the high income range

super stable poissonian structures in this paper we characterize classes of poisson processes whose statistical structures are super stable we consider a flow generated by a one dimensional ordinary differential equation and an ensemble of particles surfing the flow the particles start from random initial positions and are propagated along the flow by stochastic wave processes with general statistics and general cross correlations setting the initial positions to be poisson processes we characterize the classes of poisson processes that render the particles positions at all times and invariantly with respect to the wave processes statistically identical to their initial positions these poisson processes are termed super stable and facilitate the generalization of the notion of stationary distributions far beyond the realm of markov dynamics

random matrix theory models of electric grid topology abstract the random matrix theory is useful in the study of large systems such as electric grids these transmission systems can be modeled as complex networks with high voltage lines the edges that connect nodes representing power plants and substations we draw upon established literature of complex systems theory and introduce methods from nuclear and statistical physics to identify new characteristics of these networks we show that most grids can be characterized by the gaussian orthogonal ensemble an indicator of chaos in many complex systems under certain circumstances however grids may be described by poisson statistics an indicator of regularity we use the random matrix formalism to describe the interconnection of multiple grids and construct a simple model of a distributed grid

description of chaos order transition with random matrices within the maximum entropy principle the deformed gaussian orthogonal ensemble introduced earlier is developed here for large dimensional matrices both the spacing and eigenvector distributions are studied and compared to other ones suggested for the chaos order transition problem the concept of a universal lower entropy with respect to the gaussian orthogonal ensemble entropy is proved very useful

brownian gas models for extreme value laws in this paper we establish one dimensional brownian gas models for the extreme value laws of gumbel weibull and frechet a gas model is a countable collection of independent particles governed by common diffusion dynamics the extreme value laws are the universal probability distributions governing the affine scaling limits of the maxima and minima of ensembles of independent and identically distributed one dimensional random variables using the recently introduced concept of stationary poissonian intensities we construct two gas models whose global statistical structures are stationary and yield the extreme value laws a linear brownian motion gas model for the gumbel law and a geometric brownian motion gas model for the weibull and frechet laws the stochastic dynamics of these gas models are studied in detail and closed form analytical descriptions of their temporal correlation structures their topological phase transitions and their intrinsic first passage time fluxes are presented

approximating the largest eigenvalue of network adjacency matrices the largest eigenvalue of the adjacency matrix of a network plays an important role in several network processes e g synchronization of oscillators percolation on directed networks and linear stability of equilibria of network coupled systems in this paper we develop approximations to the largest eigenvalue of adjacency matrices and discuss the relationships between these approximations numerical experiments on simulated networks are used to test our results

ising field theory quadratic difference equations for the n point green s functions on the lattice for the two dimensional ising model at arbitrary temperature a system of quadratic difference equations is presented involving the correlations of n order variables and that of n ensuremath 2 order and two disorder variables with suitable boundary conditions these equations specify the correlations

a unified and universal explanation for levy laws and 1 f noises levy laws and 1 f noises are shown to emerge uniquely and universally from a general model of systems which superimpose the transmissions of many independent stochastic signals the signals are considered to follow statistically a common yet arbitrary generic signal pattern which may be either stationary or dissipative each signal is considered to have its own random transmission amplitude and frequency we characterize the amplitude frequency randomizations which render the system output s stationary law and power spectrum universal i e independent of the underlying generic signal pattern the classes of universal stationary laws and power spectra are shown to coincide respectively with the classes of levy laws and 1 f noises thus providing a unified and universal explanation for the ubiquity of these classes of anomalous statistics in various fields of science and engineering

handbook of the normal distribution genesis an historical background basic properties expansions and algorithms characterizations sampling distributions limit theorems and expansions normal approximations to distributions order statistics from normal samples the bivariate normal distribution bivariate normal sampling distributions point estimation statistical intervals

density and uniqueness in percolation two results on site percolation on thed dimensional lattice d 1 arbitrary are presented in the first theorem we show that for stationary underlying probability measures each infinite cluster has a well defined density with probability one the second theorem states that if in addition the probability measure satisfies the finite energy condition of newman and schulman then there can be at most one infinite cluster with probability one the simple arguments extend to a broad class of finite dimensional models including bond percolation and regular lattices

crossing probabilities for voronoi percolation we prove that the standard russo seymour welsh theory is valid for voronoi percolation this implies that at criticality the crossing probabilities for rectangles are bounded by constants depending only on their aspect ratio this result has many consequences such as the polynomial decay of the one arm event at criticality

distributed order diffusion equations and multifractality models and solutions we study distributed order time fractional diffusion equations characterized by multifractal memory kernels in contrast to the simple power law kernel of common time fractional diffusion equations based on the physical approach to anomalous diffusion provided by the seminal scher montroll weiss continuous time random walk we analyze both natural and modified form distributed order time fractional diffusion equations and compare the two approaches the mean squared displacement is obtained and its limiting behavior analyzed we derive the connection between the wiener process described by the conventional langevin equation and the dynamics encoded by the distributed order time fractional diffusion equation in terms of a generalized subordination of time a detailed analysis of the multifractal properties of distributed order diffusion equations is provided

the scaling limit of the energy correlations in non integrable ising models we obtain an explicit expression for the multipoint energy correlations of a non solvable two dimensional ising models with nearest neighbor ferromagnetic interactions plus a weak finite range interaction of strength in a scaling limit in which we send the lattice spacing to zero and the temperature to the critical one our analysis is based on an exact mapping of the model into an interacting lattice fermionic theory which generalizes the one originally used by schultz mattis and lieb for the nearest neighbor ising model the interacting model is then analyzed by a multiscale method first proposed by pinson and spencer if the lattice spacing is finite then the correlations cannot be computed in closed form rather they are expressed in terms of infinite convergent power series in in the scaling limit these infinite expansions radically simplify and reduce to the limiting energy correlations of the integrable ising model up to a finite renormalization of the parameters explicit bounds on the speed of convergence to the scaling limit are derived

spectral analysis and the dynamic response of complex networks the eigenvalues and eigenvectors of the connectivity matrix of complex networks contain information about its topology and its collective behavior in particular the spectral density ensuremath rho ensuremath lambda of this matrix reveals important network characteristics random networks follow wigner s semicircular law whereas scale free networks exhibit a triangular distribution in this paper we show that the spectral density of hierarchical networks follows a very different pattern which can be used as a fingerprint of modularity of particular importance is the value ensuremath rho 0 related to the homeostatic response of the network it is maximum for random and scale free networks but very small for hierarchical modular networks it is also large for an actual biological protein protein interaction network demonstrating that the current leading model for such networks is not adequate

statistics of atmospheric correlations for a large class of quantum systems the statistical properties of their spectrum show remarkable agreement with random matrix predictions recent advances show that the scope of random matrix theory is much wider in this work we show that the random matrix approach can be beneficially applied to a completely different classical domain namely to the empirical correlation matrices obtained from the analysis of the basic atmospheric parameters that characterize the state of atmosphere we show that the spectrum of atmospheric correlation matrices satisfy the random matrix prescription in particular the eigenmodes of the atmospheric empirical correlation matrices that have physical significance are marked by deviations from the eigenvector distribution

random graphs with arbitrary degree distributions and their applications recent work on the structure of social networks and the internet has focused attention on graphs with distributions of vertex degree that are significantly different from the poisson degree distributions that have been widely studied in the past in this paper we develop in detail the theory of random graphs with arbitrary degree distributions in addition to simple undirected unipartite graphs we examine the properties of directed and bipartite graphs among other results we derive exact expressions for the position of the phase transition at which a giant component first forms the mean component size the size of the giant component if there is one the mean number of vertices a certain distance away from a randomly chosen vertex and the average vertex vertex distance within a graph we apply our theory to some real world graphs including the worldwide web and collaboration graphs of scientists and fortune 1000 company directors we demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world while in others there is a measurable discrepancy between theory and reality perhaps indicating the presence of additional social structure in the network that is not captured by the random graph

universality in the 2d ising model and conformal invariance of fermionic observables it is widely believed that the celebrated 2d ising model at criticality has a universal and conformally invariant scaling limit which is used in deriving many of its properties however no mathematical proof has ever been given and even physics arguments support a priori weaker mobius invariance we introduce discrete holomorphic fermions for the 2d ising model at criticality on a large family of planar graphs we show that on bounded domains with appropriate boundary conditions those have universal and conformally invariant scaling limits thus proving the universality and conformal invariance conjectures

universal poisson process limits for general random walks abstract this paper considers ensembles of general independent and identically distributed random walks taking the ensemble size to grow infinitely large and also taking the running time of the random walks to grow infinitely large universal poisson process limits are obtained specifically it is established that the positions of general linear random walks converge universally to poisson processes over the real line with uniform and exponential intensities and it is established that the positions of general geometric random walks converge universally to poisson processes over the positive half line with harmonic and power intensities corollaries to these universal convergence results yield the extreme value statistics of gumbel weibull and frechet

geometric analysis of φ4 fields and ising models parts i and ii we provide here the details of the proof announced in 1 that ind 4 dimensions the even 4 euclidean field theory with a lattice cut off is inevitably free in the continuum limit in the single phase regime the analysis is nonperturbative and is based on a representation of the field variables or spins in ising systems as source sink creation operators in a system of random currents which may be viewed as the mediators of correlations in this dual representation the onset of long range order is attributed to percolation in an ensemble of sourceless currents and the physical interaction in the 4 field and other aspects of the critical behavior in ising models are directly related to the intersection properties of long current clusters an insight into the criticality of the dimensiond 4 is derived from an analogy foreseen by k symanzik with the intersection properties of paths of brownian motion other results include the proof that in certain respect the critical behavior in ising models is in exact agreement with the mean field approximation in high dimensionsd 4 but not in the low dimensiond 2 for which we establish the universality of hyperscaling

diversity of poissonian populations populations represented by collections of points scattered randomly on the real line are ubiquitous in science and engineering the statistical modeling of such populations leads naturally to poissonian populations poisson processes on the real line with a distinguished maximal point poissonian populations are infinite objects underlying key issues in statistical physics probability theory and random fractals due to their infiniteness measuring the diversity of poissonian populations depends on the lower bound cut off applied this research characterizes the classes of poissonian populations whose diversities are invariant with respect to the cut off level applied and establishes an elemental connection between these classes and extreme value theory the measures of diversity considered are variance and dispersion simpson s index and inverse participation ratio shannon s entropy and renyi s entropy and gini s index

modularity and extreme edges of the internet we study the spectral properties of a diffusion process taking place on the internet network focusing on the slowest decaying modes these modes identify an underlying modular structure roughly corresponding to individual countries for instance in the slowest decaying mode the diffusion current flows from russia to u s military sites quantitatively the modular structure manifests itself in a 10 times larger participation ratio of its slow decaying modes compared to a random scale free network we propose to use the fraction of nodes participating in slow decaying modes as a general measure of the modularity of a network for the 100 slowest decaying modes of the internet this fraction is 30 finally we suggest that the degree of isolation of an individual module can be assessed by comparing its participation in different diffusion modes

on the critical behavior of the magnetization in high dimensional ising models we derive rigorously general results on the critical behavior of the magnetization in ising models as a function of the temperature and the external field for the nearest neighbor models it is shown that ind 4 dimensions the magnetization is continuous att c and its critical exponents take the classical values 3 and 1 2 with possible logarithmic corrections atd 4 the continuity and other explicit bounds formally extend tod 3 1 2 other systems to which the results apply include long range models ind 1 dimension with 1 x y couplings for which 2 1 replacesd in the above summary the results are obtained by means of differential inequalities derived here using the random current representation which is discussed in detail for the case of a nonvanishing magnetic field

kac ward formula and its extension to order disorder correlators through a graph zeta function a streamlined derivation of the kac ward formula for the planar ising model s partition function is presented and applied in relating the kernel of the kac ward matrices inverse with the correlation functions of the ising model s order disorder correlation functions a shortcut for both is facilitated by the bowen lanford graph zeta function relation the kac ward relation is also extended here to produce a family of non planar interactions on mathbb z 2 for which the partition function and the order disorder correlators are solvable at special values of the coupling parameters temperature

record statistics of a strongly correlated time series random walks and levy flights we review recent advances on the record statistics of strongly correlated time series whose entries denote the positions of a random walk or a l evy flight on a line after a brief survey of the theory of records for independent and identically distributed random variables we focus on random walks during the last few years it was indeed realized that random walks are a very useful laboratory to test the effects of correlations on the record statistics we start with the simple one dimensional random walk with symmetric jumps both continuous and discrete and discuss in detail the statistics of the number of records as well as of the ages of the records i e the lapses of time between two successive record breaking events then we review the results that were obtained for a wide variety of random walk models including random walks with a linear drift continuous time random walks constrained random walks like the random walk bridge and the case of multiple independent random walkers finally we discuss further observables related to records like the record increments as well as some questions raised by physical applications of record statistics like the effects of measurement error and noise

fractional langevin equation we investigate fractional brownian motion with a microscopic random matrix model and introduce a fractional langevin equation we use the latter to study both subdiffusion and superdiffusion of a free particle coupled to a fractal heat bath we further compare fractional brownian motion with the fractal time process the respective mean square displacements of these two forms of anomalous diffusion exhibit the same power law behavior here we show that their lowest moments are actually all identical except the second moment of the velocity this provides a simple criterion that enable us to distinguish these two non markovian processes

langevin unification of fractional motions we present a physical langevin based theory explaining the emergence and pervasiveness of the following fractional motions brownian motion l vy motion fractional brownian motion and fractional l vy motion a general form of micro level langevin dynamics with infinitely many degrees of freedom is considered transcending from the micro level to the macro level the infinitely many degrees of freedom collapse to two characteristic exponents and the aforementioned fractional motions emerge universally the exponents categorize the fractional motions and determine their statistical and topological properties the theory establishes a unified langevin bedrock to fractional motions which are widely applied prototypical models of random transport in the sciences

randomized central limit theorems a unified theory the central limit theorems clts characterize the macroscopic statistical behavior of large ensembles of independent and identically distributed random variables the clts assert that the universal probability laws governing ensembles aggregate statistics are either gaussian or levy and that the universal probability laws governing ensembles extreme statistics are frechet weibull or gumbel the scaling schemes underlying the clts are deterministic scaling all ensemble components by a common deterministic scale however there are random environment settings in which the underlying scaling schemes are stochastic scaling the ensemble components by different random scales examples of such settings include holtsmark s law for gravitational fields and the stretched exponential law for relaxation times in this paper we establish a unified theory of randomized central limit theorems rclts in which the deterministic clt scaling schemes are replaced with stochastic scaling schemes and present randomized counterparts to the classic clts the rclt scaling schemes are shown to be governed by poisson processes with power law statistics and the rclts are shown to universally yield the levy frechet and weibull probability laws

paretian poisson processes many random populations can be modeled as a countable set of points scattered randomly on the positive half line the points may represent magnitudes of earthquakes and tornados masses of stars market values of public companies etc in this article we explore a specific class of random such populations we coin paretian poisson processes this class is elemental in statistical physics connecting together in a deep and fundamental way diverse issues including the poisson distribution of the law of small numbers paretian tail statistics the frechet distribution of extreme value theory the one sided levy distribution of the central limit theorem scale invariance renormalization and fractality resilience to random perturbations

emergent planarity in two dimensional ising models with finite range interactions the known pfaffian structure of the boundary spin correlations and more generally order disorder correlation functions is given a new explanation through simple topological considerations within the model s random current representation this perspective is then employed in the proof that the pfaffian structure of boundary correlations emerges asymptotically at criticality in ising models on mathbb z 2 with finite range interactions the analysis is enabled by new results on the stochastic geometry of the corresponding random currents the proven statement establishes an aspect of universality seen here in the emergence of fermionic structures in two dimensions beyond the solvable cases