central limit theorem for linear eigenvalue statistics of random matrices with independent entries we consider n x n real symmetric and hermitian wigner random matrices n 1 2 w with independent modulo symmetry condition entries and the null sample covariance matrices n 1 x x with independent entries of m x n matrix x assuming first that the 4th cumulant excess 4 of entries of w and x is zero and that their 4th moments satisfy a lindeberg type condition we prove that linear statistics of eigenvalues of the above matrices satisfy the central limit theorem clt as n m m n c 0 oo with the same variance as for gaussian matrices if the test functions of statistics are smooth enough essentially of the class c 5 this is done by using a simple interpolation trick from the known results for the gaussian matrices and the integration by parts presented in the form of certain differentiation formulas then by using a more elaborated version of the techniques we prove the clt in the case of nonzero excess of entries again for essentially 5 test function here the variance of statistics contains an additional term proportional to 4 the proofs of all limit theorems follow essentially the same scheme

uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory we consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn x dx e nv x dx on the line as n the potentials v are assumed to be real analytic with sufficient growth at infinity the principle results concern plancherel rotach type asymptotics for the orthogonal polynomials down to the axis using these asymptotics we then prove universality for a variety of statistical quantities arising in the theory of random matrix models some of which have been considered recently in 31 and also in 4 an additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials see also 4 the orthogonal polynomial problem is formulated as a riemann hilbert problem following 19 20 the riemann hilbert problem is analyzed in turn using the steepest descent method introduced in 12 and further developed in 11 13 a critical role in our method is played by the equilibrium measure d v for v as analyzed in 8 1999 john wiley sons inc

gaussian limit for determinantal random point fields author s soshnikov alexander abstract we prove that under fairly general conditions properly rescaled determinantal random point field converges to a generalized gaussian random process

level spacings distribution for large random matrices gaussian fluctuations we study the level spacings distribution for eigenvalues of large n x n matrices from the classical compact groups in the scaling limit when the mean distance between nearest eigenvalues equals 1 defining by 77n s the number of nearest neighbors spacings greater than s 0 smaller than s 0 we prove functional limit theorem for the process 77n s e n s n1 2 giving weak convergence of this distribution to some gaussian random process on 0 x the limiting gaussian random process is universal for all classical compact groups it is holder continuous with any exponent less than 1 2 similar results can be obtained for the n level spacings distribution

clt for linear spectral statistics of large dimensional sample covariance matrices let bn 1 n t1 2nxnx nt1 2n where xn xij is n n with i i d complex standardized entries having finite fourth moment and t1 2n is a hermitian square root of the nonnegative definite hermitian matrix tn the limiting behavior as n with n n approaching a positive constant of functionals of the eigenvalues of bn where each is given equal weight is studied due to the limiting behavior of the empirical spectral distribution of bn it is known that these linear spectral statistics converges a s to a nonrandom quantity this paper shows their rate of convergence to be 1 n by proving after proper scaling that they form a tight sequence moreover if exppx211 0 and expp x11 4 2 or if x11 and tn are real and exppx411 3 they are shown to have gaussian limits

the orbifold chow ring of toric deligne mumford stacks the orbifold chow ring of a deligne mumford stack defined by abramovich graber and vistoli 2 is the algebraic version of the orbifold cohomology ring in troduced by w chen and ruan 7 8 by design this ring incorporates numerical invariants such as the orbifold euler characteristic and the orbifold hodge num bers of the underlying variety the product structure is induced by the degree zero part of the quantum product in particular it involves gromov witten invariants inspired by string theory and results in batyrev 3 and yasuda 28 one expects that in nice situations the orbifold chow ring coincides with the chow ring of a resolution of singularities fantechi and g ttsche 14 and uribe 25 verify this conjecture when the orbifold is symn 5 where 5 is a smooth projective surface with ks 0 and the resolution is hilbn the initial motivation for this project was to compare the orbifold chow ring of a simplicial toric variety with the chow ring of a cr pant resolution to achieve this goal we first develop the theory of toric deligne mumford stacks modeled on simplicial toric varieties a toric deligne mumford stack corresponds to a combinatorial object called a stacky fan as a first approximation this object is a simplicial fan with a distinguished lattice point on each ray in the fan more precisely a stacky fan s is a triple consisting of a finitely generated abelian group n a simplicial fan e in q z n with n rays and a map zn n where the image of the standard basis in zn generates the rays in e a rational simplicial fan e produces a canonical stacky fan s n e where n is the distinguished lattice and is the map defined by the minimal lattice points on the rays hence there is a natural toric deligne mumford stack associated to every simplicial toric variety a stacky fan encodes a group action on a quasi affine variety and the toric deligne mumford stack is the quotient if e corresponds to a smooth toric variety x e and s is the canonical stacky fan associated to e then we simply have x z we show that many of the basic concepts such as open and closed toric substacks line bundles and maps between toric deligne mumford stacks correspond to combinatorial notions we expect that many more results about toric varieties lift to the realm of stacks and hope that toric deligne mumford stacks will serve as a useful testing ground for general theories

mellin barnes integrals as fourier mukai transforms we study the generalized hypergeometric system introduced by gelfand kapranov and zelevinsky and its relationship with the toric deligne mumford dm stacks recently studied by borisov chen and smith we construct series solutions with values in a combinatorial version of the chen ruan orbifold cohomology and in the k theory of the associated dm stacks in the spirit of the homological mirror symmetry conjecture of kontsevich we show that the k theory action of the fourier mukai functors associated to basic toric birational maps of dm stacks are mirrored by analytic continuation transformations of mellin barnes type

on clusterings good bad and spectral we motivate and develop a natural bicriteria measure for assessing the quality of a clustering that avoids the drawbacks of existing measures a simple recursive heuristic is shown to have poly logarithmic worst case guarantees under the new measure the main result of the article is the analysis of a popular spectral algorithm one variant of spectral clustering turns out to have effective worst case guarantees another finds a good clustering if one exists

central limit theorems for biorthogonal ensembles and asymptotics of recurrence coefficients we study fluctuations of linear statistics corresponding to smooth functions for certain biorthogonal ensembles we study those biorthogonal ensembles for which the underlying biorthogonal family s

entropy and the shannon mcmillan breiman theorem for beta random matrix ensembles we show that beta ensembles in random matrix theory with generic real analytic potential have the asymptotic equipartition property in addition we prove a central limit theorem for the density of the eigenvalues of these ensembles

normalized cuts and image segmentation we propose a novel approach for solving the perceptual grouping problem in vision rather than focusing on local features and their consistencies in the image data our approach aims at extracting the global impression of an image we treat image segmentation as a graph partitioning problem and propose a novel global criterion the normalized cut for segmenting the graph the normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups we show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion we applied this approach to segmenting static images as well as motion sequences and found the results to be very encouraging

on asymptotic behavior of multilinear eigenvalue statistics of random matrices we prove the law of large numbers and the central limit theorem for analogs of u and v von mises statistics of eigenvalues of random matrices as their size tends to infinity we show first that for a certain class of test functions kernels determining the statistics the validity of these limiting laws reduces to the validity of analogous facts for certain linear eigenvalue statistics we then check the conditions of the reduction statements for several most known ensembles of random matrices the reduction phenomenon is well known in statistics dealing with i i d random variables it is of interest that an analogous phenomenon is also the case for random matrices whose eigenvalues are strongly dependent even if the entries of matrices are independent

kronecker product graph matching abstract in this paper the interpolator based kronecker product graph matching ibkpgm algorithm for performing attributed graph matching is presented the ibkpgm algorithm is based on the kronecker product graph matching kpgm formulation this new formulation incorporates a general approach to a wide class of graph matching problems based on attributed graphs allowing the structure of the graphs to be based on multiple sets of attributes salient features of the ibkpgm algorithm are that no assumption is made about the adjacency structure of the graphs to be matched and that the explicit calculation of compatibility values between all vertices of the reference and input graphs as well as between all edges of the reference and input graphs are avoided

solution of ordinary differential equations via nonlocal transformations the classical lie analysis provides a useful technique for the solution of ordinary differential equations via point symmetries transformations unfortunately the requirement that an nth order equation possesses at least an n dimensional solvable lie algebra of symmetries is only satisfied by a select number of equations we provide a class of second order ordinary differential equations with less than the required two lie point symmetries that can be solved via nonlocal transformations

limit theorems for linear spectrum statistics of orthogonal polynomial ensembles and their applications in random matrix theory in this paper we consider the asymptotic behavior of xfn n i 1nfn xi where xi i 1 n form orthogonal polynomial ensembles and fn is a real valued bounded measurable function under the condition that varxfn n the berry esseen be bound and cramer type moderate deviation principle mdp for xfn n are obtained by using the method of cumulants as two applications we establish the be bound and cramer type mdp for linear spectrum statistics of wigner matrix and sample covariance matrix in the complex cases these results show that in the edge case which means fn has a particular form f x i x n where n is close to the right edge of equilibrium measure and f is a smooth function xfn n behaves like the eigenvalues counting function of the corresponding wigner matrix and sample covariance matrix respectively

generic behavior of the density of states in random matrix theory and equilibrium problems in the presence of real analytic external fields the equilibrium measure in the presence of an external field plays a role in a number of areas in analysis for example in random matrix theory the limiting mean density of eigenvalues is precisely the density of the equilibrium measure typical behavior for the equilibrium measure is 1 it is positive on the interior of a finite number of intervals 2 it vanishes like a square root at endpoints and 3 outside the support there is strict inequality in the euler lagrange variational conditions if these conditions hold then the limiting local eigenvalue statistics is loosely described by a bulk in which there is universal behavior involving the sine kernel and edge effects in which there is a universal behavior involving the airy kernel through techniques from potential theory and integrable systems we show that this regular behavior is generic for equilibrium measures associated with real analytic external fields in particular we show that for any one parameter family of external fields v c the equilibrium measure exhibits this regular behavior except for an at most countable number of values of c we discuss applications of our results to random matrices orthogonal polynomials and integrable systems 2000 john wiley sons inc

biextensions of picard stacks and their homological interpretation abstract let s be a site we introduce the 2 category of biextensions of strictly commutative picard s stacks we define the pull back the push down and the sum of such biextensions and we compute their homological interpretation if p q and g are strictly commutative picard s stacks the equivalence classes of biextensions of p q by g are parametrized by the cohomology group ext 1 p l q g the isomorphism classes of arrows from such a biextension to itself are parametrized by the cohomology group ext 0 p l q g and the automorphisms of an arrow from such a biextension to itself are parametrized by the cohomology group ext 1 p l q g where p q and g are the complexes associated to p q and g respectively

frobenius morphisms and derived categories on two dimensional toric deligne mumford stacks abstract for a toric deligne mumford dm stack x we can consider a certain generalization of the frobenius endomorphism for such an endomorphism f x x on a 2 dimensional toric dm stack x we show that the push forward f o x of the structure sheaf generates the bounded derived category of coherent sheaves on x we also choose a full strong exceptional collection from the set of direct summands of f o x in several examples of two dimensional toric dm orbifolds x

distribution functions for random variables for ensembles of positive hermitian matrices distribution functions for random variables that depend on a parameter are computed asymptotically for ensembles of positive hermitian matrices the inverse fourier transform of the distribution is shown to be a fredholm determinant of a certain operator that is an analogue of a wiener hopf operator the asymptotic formula shows that up to the terms of order o 1 the distributions are gaussian

the ffrt property of two dimensional normal graded rings and orbifold curves abstract we study the finite f representation type abbr ffrt property of a two dimensional normal graded ring r in characteristic p 0 using notions from the theory of algebraic stacks given a graded ring r we consider an orbifold curve c which is a root stack over the smooth curve c proj r such that r is the section ring associated with a line bundle l on c the ffrt property of r is then rephrased with respect to the frobenius push forwards f e l i on the orbifold curve c as a result we see that if the singularity of r is not log terminal then r has ffrt only in exceptional cases where the characteristic p divides a weight of c

an eigendecomposition approach to weighted graph matching problems an approximate solution to the weighted graph matching problem is discussed for both undirected and directed graphs the weighted graph matching problem is that of finding the optimum matching between two weighted graphs which are graphs with weights at each arc the proposed method uses an analytic instead of a combinatorial or iterative approach to the optimum matching problem using the eigendecompositions of the adjacency matrices in the case of the undirected graph matching problem or hermitian matrices derived from the adjacency matrices in the case of the directed graph matching problem a matching close to the optimum can be found efficiently when the graphs are sufficiently close to each other simulation results are given to evaluate the performance of the proposed method

fluctuations of linear eigenvalue statistics of β matrix models in the multi cut regime we study the asymptotic expansion in n for the partition function of matrix models with real analytic potentials in the multi cut regime up to the o n 1 terms as a result we find the limit of the generating functional of linear eigenvalue statistics and the expressions for the expectation and the variance of linear eigenvalue statistics which in the general case contain the quasi periodic in n terms

fluctuations of linear eigenvalues statistics for wigner matrices edge case in this note we consider the fluctuation theorem for x f n n sum i 1 n f lambda i i lambda i ge theta n where lambda i i 1 ldots n are eigenvalues from a wigner matrix and theta n rightarrow 2 we prove that in the edge case x f n n behaves like the counting function of wigner matrix our results can be viewed as a complement of bao et al j stat phys 150 1 88 129 2013

limiting laws of linear eigenvalue statistics for hermitian matrix models we study the variance and the laplace transform of the probability law of linear eigenvalue statistics of unitary invariant matrix models of n n hermitian matrices as n assuming that the test function of statistics is smooth enough and using the asymptotic formulas by deift et al commun pure appl math 52 1325 1425 1999 for orthogonal polynomials with varying weights we show first that if the support of the density of states of the model consists of q 2 intervals then in the global regime the variance of statistics is a quasiperiodic function of n as n generically in the potential determining the model we show next that the exponent of the laplace transform of the probability law is not in general 1 2 variance as it should be if the central limit theorem would be valid and we find the asymptotic form of the laplace transform of the probability law in certain cases

rigidity of the three dimensional hierarchical coulomb gas a random set of points in euclidean space is called rigid or hyperuniform if the number of points falling inside any given region has significantly smaller fluctuations than the corresponding number for a set of i i d random points this phenomenon has received considerable attention in recent years due to its appearance in random matrix theory the theory of coulomb gases and zeros of random analytic functions however most of the published results are in dimensions one and two this paper gives the first proof of hyperuniformity in a coulomb type system in dimension three known as the hierarchical coulomb gas this is a simplified version of the actual 3d coulomb gas the interaction potential in this model inspired by dyson s hierarchical model of the ising ferromagnet has a hierarchical structure and is locally an approximation of the coulomb potential hyperuniformity is proved at both macroscopic and microscopic scales with upper and lower bounds for the order of fluctuations that match up to logarithmic factors the fluctuations have cube root behavior in agreement with a well known prediction for the 3d coulomb gas for completeness analogous results are also proved for the 2d hierarchical coulomb gas and the 1d hierarchical log gas

eigenvalue distributions of random unitary matrices let u be an n n random matrix chosen from haar measure on the unitary group for a fixed arc of the unit circle let x be the number of eigenvalues of m which lie in the specified arc we study this random variable as the dimension n grows using the connection between toeplitz matrices and random unitary matrices and show that x e x var x 1 2 is asymptotically normally distributed in addition we show that for several fixed arcs i1 im the corresponding random variables are jointly normal in the large n limit

on the 1 n expansion for some unitary invariant ensembles of random matrices we present a version of the 1 n expansion for random matrix ensembles known as matrix models the case where the support of the density of states of an ensemble consists of one interval and the case where the density of states is even and its support consists of two symmetric intervals is treated in these cases we construct the expansion scheme for the jacobi matrix determining a large class of expectations of symmetric functions of eigenvalues of random matrices prove the asymptotic character of the scheme and give an explicit form of the first two terms this allows us in particular to clarify certain theoretical physics results on the variance of the normalized traces of the resolvent of random matrices we also find the asymptotic form of several related objects such as smoothed squares of certain orthogonal polynomials the normalized trace and the matrix elements of the resolvent of the jacobi matrices etc

limit theorems for biorthogonal ensembles and related combinatorial identities abstract we study the fluctuations of certain biorthogonal ensembles for which the underlying family p q satisfies a finite term recurrence relation of the form x p x j p x for polynomial linear statistics of such ensembles we reformulate the cumulant method introduced in 53 in terms of counting certain lattice paths on the adjacency graph of the recurrence matrix j in the spirit of 12 we show that the asymptotic fluctuations are described by the right limits of the matrix j moreover whenever the right limit is a laurent matrix we show that the clt is equivalent to soshnikov s main combinatorial lemma we discuss several applications to unitary invariant hermitian random matrices in particular we provide a general central limit theorem clt and a law of large numbers in the one cut regime we also prove a clt for the square singular values of the product of independent complex rectangular ginibre matrices as well as for the laguerre and jacobi biorthogonal ensembles introduced in 7 and we explain how to recover the equilibrium measure from the asymptotics of the recurrence coefficients finally we discuss the connection with the strong szeg limit theorem where this combinatorial method originates

first order gaussian graphs for efficient structure classification abstract first order random graphs as introduced by wong are a promising tool for structure based classification their complexity however hampers their practical application we describe an extension to first order random graphs which uses continuous gaussian distributions to model the densities of all random elements in a random graph these first order gaussian graphs foggs are shown to have several nice properties which allow for fast and efficient clustering and classification specifically we show how the entropy of a fogg may be computed directly from the gaussian parameters of its random elements this allows for fast and memoryless computation of the objective function used in the clustering procedure used for learning a graphical model of a class we give a comparative evaluation between foggs and several traditional statistical classifiers on our example problem selected from the area of document analysis our first order gaussian graph classifier significantly outperforms statistical feature based classifiers the fogg classifier achieves a classification accuracy of approximately 98 while the best statistical classifiers only manage approximately 91

on the statistical mechanics approach in the random matrix theory integrated density of states we consider the ensemble of random symmetricn n matrices specified by an orthogonal invariant probability distribution we treat this distribution as a gibbs measure of a mean field type model this allows us to show that the normalized eigenvalue counting function of this ensemble converges in probability to a nonrandom limit asn and that this limiting distribution is the solution of a certain self consistent equation

tilting generators via ample line bundles abstract it is known that a tilting generator on an algebraic variety x gives a derived equivalence between x and a certain non commutative algebra in this paper we present a method to construct a tilting generator from an ample line bundle and construct it in several examples

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

pattern vectors from algebraic graph theory graph structures have proven computationally cumbersome for pattern analysis the reason for this is that before graphs can be converted to pattern vectors correspondences must be established between the nodes of structures which are potentially of different size to overcome this problem in this paper we turn to the spectral decomposition of the laplacian matrix we show how the elements of the spectral matrix for the laplacian can be used to construct symmetric polynomials that are permutation invariants the coefficients of these polynomials can be used as graph features which can be encoded in a vectorial manner we extend this representation to graphs in which there are unary attributes on the nodes and binary attributes on the edges by using the spectral decomposition of a hermitian property matrix that can be viewed as a complex analogue of the laplacian to embed the graphs in a pattern space we explore whether the vectors of invariants can be embedded in a low dimensional space using a number of alternative strategies including principal components analysis pca multidimensional scaling mds and locality preserving projection lpp experimentally we demonstrate that the embeddings result in well defined graph clusters our experiments with the spectral representation involve both synthetic and real world data the experiments with synthetic data demonstrate that the distances between spectral feature vectors can be used to discriminate between graphs on the basis of their structure the real world experiments show that the method can be used to locate clusters of graphs

smooth toric deligne mumford stacks we give a geometric definition of smooth toric deligne mumford stacks using the action of a torus we show that our definition is equivalent to the one of borisov chen and smith in terms of stacky fans in particular we give a geometric interpretation of the combinatorial data contained in a stacky fan we also give a bottom up classification in terms of simplicial toric varieties and fiber products of root stacks

laplacian eigenmaps for dimensionality reduction and data representation one of the central problems in machine learning and pattern recognition is to develop appropriate representations for complex data we consider the problem of constructing a representation for data lying on a low dimensional manifold embedded in a high dimensional space drawing on the correspondence between the graph laplacian the laplace beltrami operator on the manifold and the connections to the heat equation we propose a geometrically motivated algorithm for representing the high dimensional data the algorithm provides a computationally efficient approach to nonlinear dimensionality reduction that has locality preserving properties and a natural connection to clustering some potential applications and illustrative examples are discussed

universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles this paper is devoted to the rigorous proof of the universality conjecture of random matrix theory according to which the limiting eigenvalue statistics ofn n random matrices within spectral intervals ofo n 1 is determined by the type of matrix real symmetric hermitian or quaternion real and by the density of states we prove this conjecture for a certain class of the hermitian matrix ensembles that arise in the quantum field theory and have the unitary invariant distribution defined by a certain function the potential in the quantum field theory satisfying some regularity conditions

incomplete determinantal processes from random matrix to poisson statistics we study linear statistics of a class of determinantal processes which interpolate between poisson and gue ginibre statistics in dimension 1 or 2 these processes are obtained by performing an independent bernoulli percolation on the particle configuration of a log gas confined in a general potential we show that depending on the expected number of deleted particles there is a universal transition for mesoscopic linear statistics namely at small scales the point process behave according to random matrix theory while at large scales it needs to be renormalized because the variance of any linear statistic diverges the crossover is explicitly characterized as the superposition of a h 1 or h 1 2 correlated gaussian noise depending on the dimension and an independent poisson process the proof consists in computing the limits of the cumulants of linear statistics using the asymptotics of the correlation kernel of the process

breakdown of universality in multi cut matrix models we solve the puzzle of the disagreement between orthogonal polynomials methods and mean field calculations for random n n matrices with a disconnected eigenvalue support we show that the difference does not stem from a 2 symmetry breaking but from the discreteness of the number of eigenvalues this leads to additional terms quasiperiodic in n which must be added to the naive mean field expressions our result invalidates the existence of a smooth topological large n expansion and some postulated universality properties of correlators we derive the large n expansion of the free energy for the general two cut case from it we rederive by a direct and easy mean field like method the two point correlators and the asymptotic orthogonal polynomials we extend our results to any number of cuts and to non real potentials

on the conjecture of king for smooth toric deligne mumford stacks abstract we construct full strong exceptional collections of line bundles on smooth toric fano deligne mumford stacks of picard number at most two and of any picard number in dimension two it is hoped that the approach of this paper will eventually lead to the proof of the existence of such collections on all smooth toric nef fano deligne mumford stacks

mock gaussian behaviour for linear statistics of classical compact groups we consider the scaling limit of linear statistics for eigenphases of a matrix taken from one of the classical compact groups we compute their moments and find that the first few moments are gaussian whereas the limiting distribution is not the precise number of gaussian moments depends upon the particular statistic considered

semiclassical asymptotics of orthogonal polynomials riemann hilbert problem and universality in the matrix model we derive semiclassical asymptotics for the orthogonal polynomials pn z on the line with respect to the exponential weight exp inv z where v z is a double well quartic polynomial in the limit when n n 1 we assume that n n cri for some 0 where cr is the critical value which separates orthogonal polynomials with two cuts from the ones with one cut simultaneously we derive semiclassical asymptotics for the recursive coe cients of the orthogonal polynomials and we show that these coe cients form a cycle of period two which drifts slowly with the change of the ratio n n the proof of the semiclassical asymptotics is based on the methods of the theory of integrable systems and on the analysis of the appropriate matrix riemannhilbert problem as an application of the semiclassical asymptotics of the orthogonal polynomials we prove the universality of the local distribution of eigenvalues in the matrix model with the double well quartic interaction in the presence of two cuts contents

shapes shocks and deformations i the components of two dimensional shape and the reaction diffusion space we undertake to develop a general theory of two dimensional shape by elucidating several principles which any such theory should meet the principles are organized around two basic intuitions first if a boundary were changed only slightly then in general its shape would change only slightly this leads us to propose an operational theory of shape based on incremental contour deformations the second intuition is that not all contours are shapes but rather only those that can enclose physical material a theory of contour deformation is derived from these principles based on abstract conservation principles and hamilton jacobi theory these principles are based on the work of sethian 1985a c the osher sethian 1988 level set formulation the classical shock theory of lax 1971 1973 as well as curve evolution theory for a curve evolving as a function of the curvature and the relation to geometric smoothing of gage hamilton grayson 1986 1989 the result is a characterization of the computational elements of shape deformations parts bends and seeds which show where to place the components of a shape the theory unifies many of the diverse aspects of shapes and leads to a space of shapes the reaction diffusion space which places shapes within a neighborhood of similar ones such similarity relationships underlie descriptions suitable for recognition

fluctuations of matrix elements of regular functions of gaussian random matrices we find the limit of the variance and prove the central limit theorem clt for the matrix elements jk m j k 1 n of a regular function of the gaussian matrix m goe and gue as its size n tends to infinity we show that unlike the linear eigenvalue statistics tr m a traditional object of random matrix theory whose variance is bounded as n and the clt is valid for tr m e tr m the variance of jk m is o 1 n and the clt is valid for sqrt n varphi jk m mathbf e varphi jk m this shows the role of eigenvectors in the forming of the asymptotic regime of various functions statistics of random matrices our proof is based on the use of the fourier transform as a basic characteristic function unlike the stieltjes transform and moments used in majority of works of the field we also comment on the validity of analogous results for other random matrices

ground states and hyperuniformity of the hierarchical coulomb gas in all dimensions stochastic point processes with coulomb interactions arise in various natural examples of statistical mechanics random matrices and optimization problems often such systems due to their natural repulsion exhibit remarkable hyperuniformity properties that is the number of points landing in any given region fluctuates at a much smaller scale compared to that of a set of i i d random points a well known conjecture from physics appearing in the works of jancovici lebowitz manificat martin and yalcin see jancovici et al in j stat phys 72 3 4 773 787 1993 lebowitz in phys rev a 27 1491 1494 1983 martin and yalcin in j stat phys 22 4 435 463 1980 states that the variance of the number of points landing in a set should grow like the surface area instead of the volume unlike i i d random points in a recent beautiful work chatterjee in rigidity of the three dimensional hierarchical coulomb gas 2017 arxiv preprint arxiv 1708 01965 to appear in probability theory and related fields chatterjee gave the first proof of such a result in dimension three for a coulomb type system known as the hierarchical coulomb gas inspired by dyson s hierarchical model of the ising ferromagnet dyson in phys rev 92 1331 1338 1953 dyson in commun math phys 12 2 91 107 1969 however the case of dimensions greater than three had remained open in this paper we establish the correct fluctuation behavior up to logarithmic factors in all dimensions greater than three for the hierarchical model using similar methods we also prove sharp variance bounds for smooth linear statistics which were unknown in any dimension bigger than two a key intermediate step is to obtain precise results about the ground states of such models whose behavior can be interpreted as hierarchical analogues of various crystalline conjectures predicted for energy minimizing systems and could be of independent interest

on the 1 n corrections to the green functions of random matrices with independent entries we propose a general approach to the construction of 1 n corrections to the green function gn z of the ensembles of random real symmetric and hermitian n n matrices with independent entries hk l by this approach we study the correlation function cn z1 z2 of the normalized trace n 1trgn assuming that the average of mod hk l5 is bounded we found that to leading order cn z1 z2 n 2f z1 z2 where f z1 2 only depends on the second and the fourth moments of hk l for the correlation function of the density of energy levels we obtain an expression which in the scaling limit only depends on the second moment of hk l this can be viewed as supporting the universality conjecture of random matrix theory

random matrix theory and ζ 1 2 it we study the characteristic polynomials z u of matrices u in the circular unitary ensemble cue of random matrix theory exact expressions for any matrix size n are derived for the moments of z and z z and from these we obtain the asymptotics of the value distributions and cumulants of the real and imaginary parts of log z as n in the limit we show that these two distributions are independent and gaussian costin and lebowitz 15 previously found the gaussian limit distribution for im log z using a different approach and our result for the cumulants proves a conjecture made by them in this case we also calculate the leading order n asymptotics of the moments of z and z z these cue results are then compared with what is known about the riemann zeta function s on its critical line re s 1 2 assuming the riemann hypothesis equating the mean density of the non trivial zeros of the zeta function at a height t up the critical line with the mean density of the matrix eigenvalues gives a connection between n and t invoking this connection our cue results coincide with a theorem of selberg for the value distribution of log 1 2 it in the limit t they are also in close agreement with numerical data computed by odlyzko 29 for large but finite t this leads us to a conjecture for the moments of 1 2 it finally we generalize our random matrix results to the circular orthogonal coe and circular symplectic cse ensembles