exact mean field solutions of the asymmetric random average process we consider the asymmetric random average process arap with continuous mass variables and parallel discrete time dynamics studied recently by krug garcia and rajesh majumdar both j statist phys 99 2000 the model is defined by an arbitrary state independent fraction density function r with support on the unit interval we examine the exactness of mean field steady state mass distributions in dependence of and identify as a conjecture based on high order calculations the class of density functions yielding product measure solutions additionally the exact form of the associated mass distributions p m is derived using these results we show examplary the exactness of the mean field ansatz for monomial fraction densities r n 1 rn 2 with n 2 for verification we calculate the mass distributions p m explicitly and prove directly that product measure holds furthermore we show that even in cases where the steady state is not given by a product measure very accurate approximants can be found in the class

distribution of infected mass in disease spreading in scale free networks we use scale free networks to study properties of the infected mass m of the network during a spreading process as a function of the infection probability q and the structural scaling exponent we use the standard sir model and investigate in detail the distribution of m we find that for dense networks this function is bimodal while for sparse networks it is a smoothly decreasing function with the distinction between the two being a function of q we thus recover the full crossover transition from one case to the other this has a result that on the same network a disease may die out immediately or persist for a considerable time depending on the initial point where it was originated thus we show that the disease evolution is significantly influenced by the structure of the underlying population

a stable indirect adaptive control scheme for first order plants with no prior knowledge on the parameters an indirect adaptive control scheme for a first order linear time invariant plant with unknown parameters is presented and analyzed the scheme requires a priori knowledge of the sign or bounds of the plant parameters a discontinuous control law involving the use of probing is utilized to avoid any singularities caused by the lack of stabilizability of the estimated model the scheme guarantees signal boundedness and zero residual tracking error

global tunability of one dimensional siso systems it is shown by example that with a suitably defined certainty equivalence controller sigma sub r it is possible to make a closed loop parameterized system sigma tunable on its entire parameter space p even though p may contain points at which the parameterized sigma sub d upon which sigma sub r s definition is based is not stabilizable the implications of this discovery are briefly discussed

an econometric model of network formation with degree heterogeneity i introduce a model of undirected dyadic link formation which allows for assortative matching on observed agent characteristics homophily as well as unrestricted agent level heterogeneity in link surplus degree heterogeneity like in fixed effects panel data analyses the joint distribution of observed and unobserved agent level characteristics is left unrestricted two estimators for the common homophily parameter 0 are developed and their properties studied under an asymptotic sequence involving a single network growing large the first tetrad logit tl estimator conditions on a sufficient statistic for the degree heterogeneity the second joint maximum likelihood jml estimator treats the degree heterogeneity a i0 i 1 super n as additional incidental parameters to be estimated the tl estimate is consistent under both sparse and dense graph sequences whereas consistency of the jml estimate is shown only under dense graph sequences

entropy maximization in the force network ensemble for granular solids a long standing issue in the area of granular media is the tail of the force distribution in particular whether this is exponential gaussian or even some other form here we resolve the issue for the case of the force network ensemble in two dimensions we demonstrate that conservation of the total area of a reciprocal tiling a direct consequence of local force balance is crucial for predicting the local stress distribution maximizing entropy while conserving the tiling area and total pressure leads to a distribution of local pressures with a generically gaussian tail that is in excellent agreement with numerics both with and without friction and for two different contact networks

contact forces in anisotropic frictional granular materials the probability density functions pdfs of contact forces in anisotropic cohesionless and frictional granular materials are studied numerically and theoretically using discrete element simulations of biaxial deformation of a large two dimensional assembly consisting of 200 000 disks it is observed that the pdfs for the normal and tangential components of the contact forces depend significantly on contact orientation the pdfs exhibit exponential decay and the pdf for the tangential component of the contact forces is not always symmetrical with respect to zero tangential force the shape of the pdf for the normal component of the contact forces changes with shear strain a qualitative explanation for this change is given that is related to the biaxial deformation mechanism in which the disrupted contacts are predominantly oriented in the direction of the minor principle stress a maximum entropy method is employed to study these pdfs theoretically using a prescribed stress tensor as constraint it is found that the theoretical results correspond qualitatively to many of the results obtained from the discrete element simulations discrepancies between theory and simulations are attributed to the fact that the kinematics have not been taken into account in the theory

models of stress fluctuations in granular media we investigate in detail two models describing how stresses propagate and fluctuate in granular media the first one is a scalar model where only the vertical component of the stress tensor is considered in the continuum limit this model is equivalent to a diffusion equation where the r ole of time is played by the vertical coordinate plus a randomly varying convection term we calculate the response and correlation function of this model and discuss several properties in particular related to the stress distribution function we then turn to the tensorial model where the basic starting point is a wave equation which in the absence of disorder leads to a ray like propagation of stress in the presence of disorder the rays acquire a diffusive width and the angle of propagation is shifted a striking feature is that the response function becomes negative which suggests that the contact network is mechanically unstable to very weak perturbations the stress correlation function reveals characteristic features related to the ray like propagation which are absent in the scalar description our analytical calculations are confirmed and extended by a numerical analysis of the stochastic wave equation

granular contact force density of states and entropy in a modified edwards ensemble a method has been found to analyze edwards granular contact force probability functional for a special case as a result the granular contact force probability density functions are obtained from first principles for this case the results are in excellent agreement with the experimental and simulation data the derivation assumes edwards flat measure a density of states dos that is uniform within the metastable regions of phase space the enabling assumption supported by physical arguments and empirical evidence is that correlating information is not significantly recursive through loops in the packing maximizing a state counting entropy results in a transport equation that can be solved numerically for the present this has been done using the mean structure approximation projecting the dos across all angular coordinates to more clearly identify its predominant nonuniformities these features are 1 the grain factor psi related to grain stability and strong correlation between the contact forces on the same grain and 2 the structure factor upsilon related to newton s third law and strong correlation between neighboring grains

simple models for granular force networks abstract a remarkable feature of static granular matter is the distribution of force along intricate networks even regular inter particle contact networks produce wildly inhomogeneous force networks where certain chains of particles carry forces far larger than the mean although the full information of each inter particle contact requires analysis of the stress tensor potentially including friction and elasticity there is a rich tradition of approximating these interactions with simple scalar models that focus on the geometry of the system in this paper we briefly review past theoretical approaches to understanding the geometry of force networks we then investigate the structure of experimentally derived granular force networks using a simple algorithm to obtain corresponding graphs we compare our observations with the results of geometric models including random bond percolation which show similar spatial distributions without enforcing vector force balance our findings suggest that some aspects of the mean geometry of granular force networks may be captured by these simple descriptions

applications of the lindeberg principle in communications and statistical learning we use a generalization of the lindeberg principle developed by s chatterjee to prove universality properties for various problems in communications statistical learning and random matrix theory we also show that these systems can be viewed as the limiting case of a properly defined sparse system the latter result is useful when the sparse systems are easier to analyze than their dense counterparts the list of problems we consider is by no means exhaustive we believe that the ideas can be used in many other problems relevant for information theory

a deterministic equivalent for the analysis of non gaussian correlated mimo multiple access channels using large dimensional random matrix theory rmt we conduct mutual information analysis of a multiple input multiple output mimo multiple access channel mac our channel model reflects the characteristics in small cell networks where antenna correlations line of sight components and general type of fading distributions have to be included the mutual information expression can be expressed as functionals of the stieltjes transform through the so called shannon transform ideally if the stieltjes transform is known in the context of the large dimensional rmt then the problem is solved however it is difficult to derive the stieltjes transform of the considered channel models directly especially when the transmit correlation matrices are generally nonnegative definite and the channel entries are non gaussian to overcome this we use the generalized lindeberg principle to show that the stieltjes transforms of this class of random matrices with gaussian or non gaussian independent entries coincide in the large dimensional regime this result permits to derive the deterministic equivalents e g the stieltjes transform and the ergodic mutual information for non gaussian mimo channels from the known results developed for gaussian mimo channels as an application we determine the capacity achieving input covariance matrices for the mimo macs and prove that the capacity achieving input covariance matrices are asymptotically independent of the fading distribution

a law of large numbers for finite range dependent random matrices we consider random hermitian matrices in which distant above diagonal entries are independent but nearby entries may be correlated we find the limit of the empirical distribution of eigenvalues by combinatorial methods we also prove that the limit has an algebraic stieltjes transform by an argument based on dimension theory of noetherian local rings 2008 wiley periodicals inc

forces on bins the effect of random friction the q model of coppersmith et al phys rev e 53 4673 1996 has renewed interest in understanding the forces generated along the walls and at the bottom of a silo filled with a granular material fluctuations in the mean stress have been characterized for the q model and related to experimental work on stress chains the classical engineering approach to bin loads follows from janssen s analysis z ver dtsch ing 39 1045 1895 predicting a saturation of stress as a function of depth in a tall silo in this paper we reexamine the janssen theory introducing randomness into the important parameters in the theory the janssen analysis relies on assumptions not met in practice for this reason we numerically solve the partial differential equations governing the equilibrium of forces in a bin again including randomness in parameters we show that the most important of these parameters is a coefficient of friction at the wall of the bin this random friction model combines some features of fluctuations as seen in experiments with a classical continuum mechanics approach to describing granular materials

probability density functions of contact forces for cohesionless frictional granular materials a theory is developed for the probability density functions of contact forces for cohesionless frictional granular materials in quasi static equilibrium this theory is based on a maximum information entropy principle with an expression for information entropy that is appropriate for granular materials entropy is maximized under the constraints of a prescribed stress and that the normal component of the contact force is compressive and that the tangential component of the contact force is limited by coulomb friction the theory results in a dependence of the probability density function for the tangential contact forces on the friction coefficient the theoretical predictions are compared with results from discrete element simulations on isotropic two dimensional assemblies under hydrostatic stress good qualitative agreement is found for means and standard deviations of contact forces and the shape of the probability density functions while the quantitative agreement is fairly good discrepancies between theory and simulations such as the difference in shape of the probability density function for the normal force and the observed dependence on elastic properties of the exponential decay rate of tangential forces are attributed to the fact that the method does not take into account any kinematics which are essential in relation to elastic effects

model for force fluctuations in bead packs we study theoretically the complex network of forces that is responsible for the static structure and properties of granular materials we present detailed calculations for a model in which the fluctuations in the force distribution arise because of variations in the contact angles and the constraints imposed by the force balance on each bead of the pile we compare our results for the force distribution function for this model including exact results for certain contact angle probability distributions with numerical simulations of force distributions in random sphere packings this model reproduces many aspects of the force distribution observed both in experiment and in numerical simulations of sphere packings our model is closely related to some that have been studied in the context of self organized criticality we present evidence that in the force distribution context critical power law force distributions occur only when a parameter hidden in other interpretations is tuned our numerical mean field and exact results all indicate that for almost all contact distributions the distribution of forces decays exponentially at large forces

the isotropic semicircle law and deformation of wigner matrices we analyze the spectrum of additive finite rank deformations of n n wigner matrices h the spectrum of the deformed matrix undergoes a transition associated with the creation or annihilation of an outlier when an eigenvalue di of the deformation crosses a critical value 1 this transition happens on the scale we allow the eigenvalues di of the deformation to depend on n under the condition we make no assumptions on the eigenvectors of the deformation in the limit n we identify the law of the outliers and prove that the nonoutliers close to the spectral edge have a universal distribution coinciding with that of the extremal eigenvalues of a gaussian matrix ensemble a key ingredient in our proof is the isotropic local semicircle law which establishes optimal high probability bounds on where m z is the stieltjes transform of wigner s semicircle law and v w are arbitrary deterministic vectors 2013 wiley periodicals inc

the missing stress geometry equation in granular media abstract the simplest solvable problem of stress transmission through a static granular material is when the grains are perfectly rigid and have an average coordination number of z d 1 under these conditions there exists an analysis of stress which is independent of the analysis of strain and the d equations of force balance j ij r g i r have to be supported by d d 1 2 equations these equations are of purely geometric origin a method of deriving them has been proposed in an earlier paper edwards and grinev phys rev lett 82 1999 5397 in this paper alternative derivations are discussed and the problem of the missing equations is posed as a geometrical puzzle which has yet to find a systematic solution as against sensible but fundamentally arbitrary approaches

on distribution of contact forces in random granular packings abstract in an earlier paper ngan phys rev e 68 2003 011301 proposed a statistical mechanics based theory to predict the mathematical form of the probability force distribution of contact forces in stressed granular packings however the wrong choice of the mean force constraint in that work leads to unsatisfactory prediction in the 3d situation although the 2d prediction is very accurate the present work shows that a different mean strain constraint chosen for the force distribution can yield the correct prediction when compared to computer simulation results in both 2d and 3d furthermore new simulation results are used to verify the interpretation that the shape of the force distribution measured by a parameter in the predicted force distribution is a structure sensitive term

applications of hysteresis switching in parameter adaptive control the hysteresis switching algorithm of r h middleton et al ibid vol 33 no 1 p 50 8 jan 1988 is reexamined in a broader context to demonstrate its utility the algorithm is applied to various families of identifier based parameterized controllers of both the direct and indirect control types application to the direct control type results in a model reference adaptive controller capable of stabilizing without excitation any siso process which can be modeled by a minimum phase linear system whose transfer function has relative and mcmillan degrees not exceeding prescribing integers m and n respectively it is shown that such processes can also be adaptively stabilized with indirect adaptive controllers and hysteresis switching a simple numerical example involving a non minimum phase process model is used to illustrate how hysteresis switching might be applied to implicitly tuned parameterized controllers to realize an adaptive controller with capabilities which might prove very difficult if not impossible to achieve without hysteresis switching or some other form of discontinuous control

approximation to stable law by the lindeberg principle abstract by the lindeberg principle we develop in this paper an approximation to one dimensional possibly asymmetric stable distributions with 0 2 in the smooth wasserstein distance it is the first time that the general stable central limit theorem is proved by the lindeberg principle and that this theorem with 0 1 is proved by a new method other than fourier analysis our main tools are a taylor like expansion and a kolmogorov forward equation

central limit theorems and bootstrap in high dimensions this paper derives central limit and bootstrap theorems for probabilities that sums of centered high dimensional random vectors hit hyperrectangles and sparsely convex sets specifically we derive gaussian and bootstrap approximations for probabilities p n 1 2 ni 1xi a p n 1 2 i 1nxi a where x1 xnx1 xn are independent random vectors in rprp and aa is a hyperrectangle or more generally a sparsely convex set and show that the approximation error converges to zero even if p pn p pn as n n and p np n in particular pp can be as large as o ecnc o ecnc for some constants c c 0c c 0 the result holds uniformly over all hyperrectangles or more generally sparsely convex sets and does not require any restriction on the correlation structure among coordinates of xixi sparsely convex sets are sets that can be represented as intersections of many convex sets whose indicator functions depend only on a small subset of their arguments with hyperrectangles being a special case

edge universality of correlation matrices xij 0 and e e 2 1 m n m these entries have a sub exponential decay at the tails we will be working in the regime n m dn limn 1 dn 6 0 1 in this paper we prove the edge universality of correlation matrices x x where the rectangular matrix x called the standardized matrix is obtained by normalizing each column of the data matrix e x by its euclidean norm our main result states that asymptotically the k point k 1 correlation functions of the extreme eigenvalues at both edges of the spectrum of the correlation matrix x x converge to those of the gaussian correlation matrix i e tracy widom law and thus in particular the largest and the smallest eigenvalues of x x after appropriate centering and rescaling converge to the tracy widom distribution the asymptotic distribution of extreme eigenvalues of the gaussian correlation matrix has been worked out only recently as a corollary of the main result in this paper we also obtain that the extreme eigenvalues of gaussian correlation matrices are asymptotically distributed according to the tracywidom law the proof is based on the comparison of green functions but the key obstacle to be surmounted is the strong dependence of the entries of the correlation matrix we achieve this via a novel argument which involves comparing the moments of product of the entries of the standardized data matrix to those of the raw data matrix our proof strategy may be extended for proving the edge universality of other random matrix ensembles with dependent entries and hence is of independent interest

asymptotic behavior of a stationary silo with absorbing walls we study the nearest neighbors one dimensional uniform q model of force fluctuations in bead packs 1 a stochastic model to simulate the stress of granular media in two dimensional silos the vertical coordinate plays the role of time and the horizontal coordinate the role of space the process is a discrete time markov process with state space 1 n at each layer time the weight supported by each grain is a random variable of mean one its own weight plus the sum of random fractions of the weights supported by the nearest neighboring grains at the previous layer the fraction of the weight given to the right neighbor of the successive layer is a uniform random variable in 0 1 independent of everything the remaining weight is given to the left neighbor in the boundaries a uniform fraction of the weight leans on the wall of the silo this corresponds to absorbing boundary conditions for this model we show that there exists a unique invariant measure the mean weight at site i under the invariant measure is i n 1 i we prove that its variance is frac 1 2 i n 1 i 2 o n3 and the covariances between grains i j are of order o n3 moreover as n the law under the invariant measure of the weights divided by n2 around site integer part of rn r 0 1 converges to a product of gamma distributions with parameters 2 and 2 r 1 r 1 sum of two exponentials of mean r 1 r 2 liu et al 2 proved that for a silo with infinitely many weightless grains any product of gamma distributions with parameters 2 and 2 with 0 are invariant our result shows that as the silo grows the model selects exactly one of these gamma s at each macroscopic place

average stresses and force fluctuations in noncohesive granular materials a lattice model is presented for investigating the fluctuations in static granular materials under gravitationally induced stress the model is similar in spirit to the scalar q model of coppersmith et al phys rev e 53 4673 1996 but ensures balance of all components of forces and torques at each site the geometric randomness in real granular materials is modeled by choosing random variables at each site consistent with the assumption of cohesionless grains configurations of the model can be generated rapidly allowing the statistical study of relatively large systems for a two dimensional 2d system with rough walls the model generates configurations consistent with continuum theories for the average stresses unlike the q model without requiring the assumption of a constitutive relation for a 2d system with periodic boundary conditions the model generates single grain force distributions similar to those obtained from the q model with a singular distribution of q s

scheidegger s rivers takayasu s aggregates and continued fractions abstract a discrete model of 1 1 dimensional aggregation equivalent to scheidegger s 1967 river model is tackled with continued fractions an exact expression for the power law distribution of aggregate masses is found

a generalization of the lindeberg principle we generalize lindeberg s proof of the central limit theorem to an invariance principle for arbitrary smooth functions of independent and weakly dependent random variables the result is applied to get a similar theorem for smooth functions of exchangeable random variables this theorem allows us to identify for the first time the limiting spectral distributions of wigner matrices with exchangeable entries

anisotropy in granular media classical elasticity and directed force chain network a general approach is presented for understanding the stress response function in anisotropic granular layers in two dimensions the formalism accommodates both classical anisotropic elasticity theory and linear theories of anisotropic directed force chain networks perhaps surprisingly two peak response functions can occur even for classical anisotropic elastic materials such as triangular networks of springs with different stiffnesses in such cases the peak widths grow linearly with the height of the layer contrary to the diffusive spreading found in stress only hyperbolic models in principle directed force chain networks can exhibit the two peak diffusively spreading response function of hyperbolic models but all models in a particular class studied here are found to be in the elliptic regime

a remark on a theorem of chatterjee and last passage percolation in this paper we prove universality of random matrix fluctuations of the last passage time of last passage percolation lpp in thin rectangles the proof is a simple corollary of a theorem of chatterjee this gives an alternative and more elementary proof of the same result in baik and suidan 2005 int math res not 325 37 and bodineau and martin 2005 electron commun probab 10 105 12 electronic in the special case of finite third moment

fractional diffusion model for force distribution in static granular media we present the results of a numerical and an experimental investigation of the probability distribution of normal contact forces in static packs of particles with two different hardnesses force distributions are computed and compared with existing models and experimental data it is found that the probability distribution function of normal contact forces p f is well described by a semiempirical model derived from a fractional diffusion equation this model reproduces most of the features common to force distributions observed in experimental and numerical studies including the finite value for p f as the forces tend to zero the results indicate that the fractional model fits well both the numerical and experimental data over a wide range of particle deformations in contrast to the existing models these results provide an insight into the physics of granular media and complement previous findings

fundamental limits of symmetric low rank matrix estimation we consider the high dimensional inference problem where the signal is a low rank symmetric matrix which is corrupted by an additive gaussian noise given a probabilistic model for the low rank matrix we compute the limit in the large dimension setting for the mutual information between the signal and the observations as well as the matrix minimum mean squared error while the rank of the signal remains constant we also show that our model extends beyond the particular case of additive gaussian noise and we prove an universality result connecting the community detection problem to our gaussian framework we unify and generalize a number of recent works on pca sparse pca submatrix localization or community detection by computing the information theoretic limits for these problems in the high noise regime in addition we show that the posterior distribution of the signal given the observations is characterized by a parameter of the same dimension as the square of the rank of the signal i e scalar in the case of rank one this allows to locate precisely the information theoretic thresholds for the above mentioned problems finally we connect our work with the hard but detectable conjecture in statistical physics

on the marcenko pastur law for linear time series institute of mathematical statistics 2015 this paper is concerned with extensions of the classical ma enko pastur law to time series specifically p dimensional linear processes are considered which are built from innovation vectors with independent identically distributed real or complex valued entries possessing zero mean unit variance and finite fourth moments the coefficient matrices of the linear process are assumed to be simultaneously diagonalizable in this setting the limiting behavior of the empirical spectral distribution of both sample covariance and symmetrized sample autocovariance matrices is determined in the high dimensional setting p n c 0 for which dimension p and sample size n diverge to infinity at the same rate the results extend existing contributions available in the literature for the covariance case and are one of the first of their kind for the autocovariance case

exactly solved model of self organized critical phenomena we define a variant of the model of bak tang and wiesenfeld of self organized critial behavior by introducing a preferred direction we characterize the critical state and by establishing equivalence to a voter model determine the critical exponents exactly in arbitrary dimension d the upper critical dimension for this model is three in two dimensions the model is equivalent to an earlier solved special case of directed percolation

a cyclic switching strategy for parameter adaptive control this paper introduces a new strategy called cyclic switching to deal with the well known certainty equivalence control synthesis problem which arises in the design of identifier based adaptive controllers because of the existence of points in parameter space where the design model spl sigma sub d upon which certainty equivalence synthesis is based loses stabilizability unlike most previously suggested methods for handling this problem the technique proposed here can be employed with or without process excitation for the technique to work it is not necessary for there to be a mechanism for moving tuned parameters away from values at which spl sigma sub d loses stabilizability and no such mechanism is used

towards a unified theory of parameter adaptive control ii certainty equivalence and implicit tuning for pt i see ibid vol 35 p 1002 12 1990 it is shown that for a large class of linear multivariable process models a properly designed certainty equivalence controller results in a tunable closed loop parameterized system the importance of this result to both the analysis and synthesis of parameter adaptive control systems is discussed it is demonstrated by means of examples how the connection between certainty equivalence and tunability together with the concept of tunability itself can be used to markedly simplify the stability analysis of adaptive control systems a new family of indirect parameterized controllers is introduced which has capabilities comparable to those of the well known direct parameterized controllers which have served as basic building blocks for adaptive control systems of all types the concept of implicit tuning is formalized and its potential importance to adaptive control is briefly discussed it is explained how the idea of tunability can be used to analyze an implicitly tuned system

anisotropic local laws for random matrices we develop a new method for deriving local laws for a large class of random matrices it is applicable to many matrix models built from sums and products of deterministic or independent random matrices in particular it may be used to obtain local laws for matrix ensembles that are anisotropic in the sense that their resolvents are well approximated by deterministic matrices that are not multiples of the identity for definiteness we present the method for sample covariance matrices of the form open image in new window where t is deterministic and x is random with independent entries we prove that with high probability the resolvent of q is close to a deterministic matrix with an optimal error bound and down to optimal spectral scales as an application we prove the edge universality of q by establishing the tracy widom airy statistics of the eigenvalues of q near the soft edges this result applies in the single cut and multi cut cases further applications include the distribution of the eigenvectors and an analysis of the outliers and bbp type phase transitions in finite rank deformations they will appear elsewhere we also apply our method to wigner matrices whose entries have arbitrary expectation i e we consider w a where w is a wigner matrix and a a hermitian deterministic matrix we prove the anisotropic local law for w a and use it to establish edge universality

asymptotic properties of large random matrices with independent entries we study the normalized trace gn z n 1 tr h zi 1 of the resolvent of n n real symmetric matrices h 1 jk wjk n j k 1n assuming that their entries are independent but not necessarily identically distributed random variables we develop a rigorous method of asymptotic analysis of moments of gn z for iz 0 where 0 is determined by the second moment of wjk by using this method we find the asymptotic form of the expectation e gn z and of the connected correlator e gn z1 gn z2 e gn z1 e gn z2 we also prove that the centralized trace ngn z e ngn z has the gaussian distribution in the limit n based on these results we present heuristic arguments supporting the universality property of the local eigenvalue statistics for this class of random matrix ensembles

sparse regular random graphs spectral density and eigenvectors we examine the empirical distribution of the eigenvalues and the eigenvectors of adjacency matrices of sparse regular random graphs we find that when the degree sequence of the graph slowly increases to infinity with the number of vertices the empirical spectral distribution converges to the semicircle law moreover we prove concentration estimates on the number of eigenvalues over progressively smaller intervals we also show that with high probability all the eigenvectors are delocalized