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

universality at the edge of the spectrum for unitary orthogonal and symplectic ensembles of random matrices we prove universality at the edge of the spectrum for unitary 2 orthogonal 1 and symplectic 4 ensembles of random matrices in the scaling limit for a class of weights w x e v x where v is a polynomial v x 2mx2m 2m 0 the precise statement of our results is given in theorem 1 1 and corollaries 1 2 and 1 4 below for the same class of weights a proof of universality in the bulk of the spectrum is given in 12 for the unitary ensembles and in 9 for the orthogonal and symplectic ensembles our starting point in the unitary case is 12 and for the orthogonal and symplectic cases we rely on our recent work 9 which in turn depends on the earlier work of widom 46 and tracy and widom 42 as in 9 the uniform plancherel rotach type asymptotics for the orthogonal polynomials found in 12 plays a central role the formulae in 46 express the correlation kernels for 1 4 as a sum of a christoffel darboux cd term as in the case 2 together with a correction term in the bulk scaling limit 9 the correction term is of lower order and does not contribute to the limiting form of the correlation kernel by contrast in the edge scaling limit considered here the cd term and the correction term contribute to the same order this leads to additional technical difficulties over and above 49 2006 wiley periodicals inc

the small dispersion limit of the korteweg de vries equation iii in parts i and ii we have derived explicit formulas for the distribution limit u of the solution of the kdv equation as the coefficient of uxxx tends to zero this formula contains n parameters 1 n whose values as well as whose number depends on x and t in section 4 we have shown that for t tb n 1 and the value of was determined in section 5 we have shown that the parameters i satisfy a nonlinear system of partial differential equations in part iii section 6 we show that for t large n 3 and we determine the asymptotic behavior of 1 2 3 and of u and u2 for t large the explicit formulas show that u and u2 are o t 1 and o t 2 respectively see formulas 6 2 and 6 24 in section 7 we study initial data whose value tends to zero as x and to 1 as x if one accepts some plausible guesses about the behavior of solutions with such initial data we derive an explicit formula for the solution and determine the large scale asymptotic behavior of the solution the function s is expressible in terms of complete elliptic integrals a similar formula is derived for u2 in section 8 we indicate how to extend the treatment of this series of papers to multihumped but still negative initial data

mesoscopic colonization of a spectral band we consider the unitary matrix model in the limit where the size of the matrices becomes infinite and in the critical situation when a new spectral band is about to emerge in previous works the number of expected eigenvalues in the neighborhood of the band was fixed and finite a situation that was termed birth of a cut or first colonization we now consider the transitional regime where this microscopic population in the new band grows without bounds but at a slower rate than the size of the matrix the local population in the new band organizes in a mesoscopic regime in between the macroscopic behavior of the full system and the previously studied microscopic one the mesoscopic colony may form a finite number of new bands with a maximum number dictated by the degree of criticality of the original potential we describe the delicate scaling limit that realizes and controls the mesoscopic colony the method we use is the steepest descent analysis of the riemann hilbert problem that is satisfied by the associated orthogonal polynomials

strong asymptotics of orthogonal polynomials with respect to exponential weights we consider asymptotics of orthogonal polynomials with respect to weights w x dx e q x dx on the real line where q x 2m k 0 qkx k q2m 0 denotes a polynomial of even order with positive leading coefficient the orthogonal polynomial problem is formulated as a riemann hilbert problem following 22 23 we employ the steepest descent type method introduced in 18 and further developed in 17 19 in order to obtain uniform plancherel rotach type asymptotics in the entire complex plane as well as asymptotic formulae for the zeros the leading coefficients and the recurrence coefficients of the orthogonal polynomials c 1999 john wiley sons inc

transport maps for beta matrix models and universality we construct approximate transport maps for non critical beta matrix models that is maps so that the push forward of a non critical beta matrix model with a given potential is a non critical beta matrix model with another potential up to a small error in the total variation distance one of the main features of our construction is that these maps enjoy regularity estimates that are uniform in the dimension in addition we find a very useful asymptotic expansion for such maps which allows us to deduce that local statistics have the same asymptotic behavior for both models

the isomonodromy approach to matric models in 2d quantum gravity we consider the double scaling limit in the hermitian matrix model for 2d quantum gravity associated with the measure exp sum limits j 1 n t j z 2j n geqq 3 we show that after the appropriate modification of the contour of integration the cross migdal douglas shenker limit to the painleve i equation in the generic case of the pure gravity is valid and calculate the nonperturbative parameters of the corresponding painleve function our approach is based on the wkb analysis of the l a pair corresponding to the discrete string equation in the framework of the inverse monodromy method here we extend our results which were obtained before for the particular casesn 2 3 our analysis complements the isomonodromy approach proposed by g moore to the general string equations that come from the matrix model in the continuous limit and differ in that we apply the isomonodromy technique to investigate the double scaling limit itself

ant colony optimization based binary search for efficient point pattern matching in images abstract point pattern matching ppm is a task to pair up the points in two images of a same scene there are many existing approaches in literature for point pattern matching however the drawback lies in the high complexity of the algorithms to overcome this drawback an ant colony optimization based binary search point pattern matching acobsppm algorithm is proposed according to this approach the edges of the image are stored in the form of point patterns to match an incoming image with the stored images the ant agent chooses a point value in the incoming image point pattern and employs a binary search method to find a match with the point values in the stored image point pattern chosen for comparison once a match occurs the ant agent finds a match for the next point value in the incoming image point pattern by searching between the matching position and maximum number of point values in the stored image point pattern the stored image point pattern having the maximum number of matches is the image matching with the incoming image experimental results are shown to prove that acobsppm algorithm is efficient when compared to the existing point pattern matching approaches in terms of time complexity and precision accuracy

the asymptotics of monotone subsequences of involutions we compute the limiting distributions of the lengths of the longest monotone subsequences of random signed involutions with or without conditions on the number of fixed points and negated points as the sizes of the involutions tend to infinity the resulting distributions are depending on the number of fixed points 1 the tracy widom distributions for the largest eigenvalues of random goe gue gse matrices 2 the normal distribution or 3 new classes of distributions which interpolate between pairs of the tracy widom distributions we also consider the second rows of the corresponding young diagrams in each case the convergence of moments is also shown the proof is based on the algebraic work of j baik and e rains in 7 which establishes a connection between the statistics of random involutions and a family of orthogonal polynomials and an asymptotic analysis of the orthogonal polynomials which is obtained by extending the riemann hilbert analysis for the orthogonal polynomials by p deift k johansson and baik in 3

sum rules via large deviations in this paper a sum rule means a relationship between a functional defined on a subset of all probability measures on r involving the reverse kullback leibler divergence with respect to a particular distribution and recursion coefficients related to the orthogonal polynomial construction the first sum rule is the szeg o verblunsky theorem see simon 2011 theorem 1 8 6 p 29 and concerns the case of trigonometrical polynomials on the torus more recently killip and simon killip and simon 2003 have given a revival interest to this subject by showing a quite surprising sum rule for measures dominating the semicircular distribution indeed this sum rule includes a contribution of the atomic part of the measure away from the support of the circular law in this paper we recover this sum rules by using only large deviations tools on random matrices furthermore we obtain new up to our knowledge magic sum rules for the reverse kullback leibler divergence with respect to the pastur marchenko or kesten mckay distributions as in the semicircular case these formulas generally include a contribution of the atomic part appearing away from the support of the reference measure from the point of view of large deviations we obtain these formulas by showing a large deviation principle for the point mass measure concentrated on random eigenvalues of classical ensembles and combine it with a large deviation principle due to gamboa and rouault 2011 for the recursion coefficients

discrete painleve equations and their appearance in quantum gravity we discuss an algorithmic approach for both deriving discrete analogues of painleve equations as well as using such equations to characterize similarity reductions of spatially discrete integrable evolution equations as a concrete example we show that a discrete analogue of painleve i can be used to characterize similarity solutions of the kac moerbeke equation it turns out that these similarity solutions also satisfy a special case of painleve iv equation in addition we discuss a methodology for obtaining the relevant continuous limits not only at the level of equations but also at the level of solutions as an example we use the wkb method in the presence of two turning points of the third order to parametrize at the continuous limit the solution of painleve i in terms of the solution of discrete painleve i finally we show that these results are useful for investigating the partition function of the matrix model in 2d quantum gravity associated with the measure exp t1z2 t2z4 t3z6

a fast expected time algorithm for the 2 d point pattern matching problem point set pattern matching is an integral part ofmany pattern recognition problems we study a randomized algorithm f or the alignment approach to model based recognition under certain mild assumptions we show that ifour scene is a set of n points and our model is a set of min points our algorithm has expected running time o n log m 3 2 for 5nding an occurrence of the model in the scene this is signi5cantly faster than any existing algorithms in the literature we then describe some experimental results on randomly generated data using a practical version ofour algorithm these results agree well with the theoretical analysis 2004 pattern recognition society published by elsevier ltd all rights reserved

extreme gaps between eigenvalues of random matrices this paper studies the extreme gaps between eigenvalues of random matrices we give the joint limiting law of the smallest gaps for haar distributed unitary matrices and matrices from the gaussian unitary ensemble in particular the kth smallest gap normalized by a factor n 4 3 has a limiting density proportional to x3k 1e x3 concerning the largest gaps normalized by n logn they converge in lp to a constant for all p 0 these results are compared with the extreme gaps between zeros of the riemann zeta function

topological recursion in enumerative geometry and random matrices we review the method of symplectic invariants recently introduced to solve matrix models loop equations in the so called topological expansion and further extended beyond the context of matrix models for any given spectral curve one defines a sequence of differential forms and a sequence of complex numbers fg called symplectic invariants we recall the definition of fg s and we explain their main properties in particular symplectic invariance integrability modularity as well as their limits and their deformations then we give several examples of applications in particular matrix models enumeration of discrete surfaces maps algebraic geometry and topological strings and non intersecting brownian motions

the small dispersion limit of the korteweg de vries equation ii in part i the scattering transform method is used to study the weak limit of solutions to the initial value problem for the korteweg devries kdv equation as the dispersion tends to zero in that limit the associated schrodinger operator becomes semiclassical so the exact scattering data is replaced by its corresponding wkb expressions only nonpositive initial data are considered in that case the limiting reflection coefficient vanishes the explicit solution of kay and moses for the reflectionless inverse transform is then analyzed and the weak limit valid for all time is characterized by a quadratic minimum problem with constraints this minimum problem is reduced to a riemann hilbert problem in function theory in parts ii and iii we use function theoretical methods to solve the riemann hilbert problem in terms of solutions to an auxiliary initial value problem the weak limit satisfies the kdv equation with the dispersive term dropped until its derivatives become infinite up to that time the weak limit is a strong l2 limit at later times the weak limit is locally described by whitham s averaged equations or more generally by the equations found by flaschka et al using multiphase averaging for large times behavior of the weak limit is studied directly from the minimum problem

gaussian fluctuation for the number of particles in airy bessel sine and other determinantal random point fields we prove the central limit theorem clt for the number of eigenvalues near the spectrum edge for certain hermitian ensembles of random matrices to derive our results we use a general theorem essentially due to costin and lebowitz concerning the gaussian fluctuation of the number of particles in random point fields with determinantal correlation functions as another corollary of the costin lebowitz theorem we prove the clt for the empirical distribution function of the eigenvalues of random matrices from classical compact groups

strong asymptotics of the orthogonal polynomials with respect to a measure supported on the plane we consider the orthogonal polynomials pn z with respect to the measure z a 2nce n z 2da z over the whole complex plane we obtain the strong asymptotic of the orthogonal polynomials in the complex plane and the location of their zeros in a scaling limit where n grows to infinity with n the asymptotics are described in terms of three probability measures associated with the problem the first measure is the limit of the counting measure of zeros of the polynomials which is captured by the g function much in the spirit of ordinary orthogonal polynomials on the real line the second measure is the equilibrium measure that minimizes a certain logarithmic potential energy supported on a region k of the complex plane the third measure is the harmonic measure of kc with a pole at this appears as the limit of the probability measure given up to the normalization constant by the squared modulus of the nth orthogonal polynomial times the orthogonality measure i e pn z 2 z a 2nce n z 2da z the compact region k that is the support of the second measure undergoes a topological transition under the variation of the parameter t n n in a double scaling limit near the critical point given by tc a a 2c we observe the hastings mcleod solution to painleve ii in the asymptotics of the orthogonal polynomials 2014 wiley periodicals inc

the behavior of the weyl function in the zero dispersion kdv limit the moment formulas that globally characterize the zero dispersion limit of the korteweg devries kdv equation are known to be expressed in terms of the solution of a maximization problem here we establish a direct relation between this maximizer and the zero dispersion limit of the logarithm of the jost functions associated with the inverse spectral transform all the kdv conserved densities are encoded in the spatial derivative of these functions known as weyl functions we show the weyl functions are densities of measures that converge in the weak sense to a limiting measure this limiting measure encodes all of the weak limits of the kdv conserved densities moreover we establish the weak limit of spectral measures associated with the dirichlet problem

approximate input sensitive algorithms for point pattern matching we study input sensitive algorithms for point pattern matching under various transformations and the hausdorff metric as a distance function given point sets p and q in the plane the problem of point pattern matching is to determine whether p is similar to some portion of q where p may undergo transformations from a group g of allowed transformations all algorithms are based on methods for extracting small subsets from q that can be matched to a small subset of p the runtime is proportional to the number k of these subsets let d be the number of points in p that are needed to define a transformation in g the key observation is that for some set b p of cardinality larger than d the number of subsets of q of this cardinality that match b is practically small as the problem becomes more constrained we present methods to extract efficiently all these subsets in q we provide algorithms for homothetic rigid and similarity transformations in the plane and give a general method that works for any dimension and for any group of transformations the runtime of our algorithms depends roughly linearly on the number of subsets k in addition to an nlogn factor thus our approximate matching algorithms run roughly in time o nlogn kmlogn where m and n are the number of points in p and q respectively the constants hidden in the big o vary depending on the group of transformations g

harmonic analysis in phase space and finite weyl heisenberg ensembles weyl heisenberg ensembles are translation invariant determinantal point processes on mathbb r 2d associated with the schrodinger representation of the heisenberg group and include as examples the ginibre ensemble and the polyanalytic ensembles which model the higher landau levels in physics we introduce finite versions of the weyl heisenberg ensembles and show that they behave analogously to the finite ginibre ensembles more specifically guided by the observation that the ginibre ensemble with n points is asymptotically close to the restriction of the infinite ginibre ensemble to the disk of area n we define finite wh ensembles as adequate finite approximations of the restriction of infinite wh ensembles to a given domain omega we provide a precise rate for the convergence of the corresponding one point intensities to the indicator function of omega as omega is dilated and the process is rescaled proportionally thermodynamic regime the construction and analysis rely neither on explicit formulas nor on the asymptotics for orthogonal polynomials but rather on phase space methods second we apply our construction to study the pure finite ginibre type polyanalytic ensembles which model finite particle systems in a single landau level and are defined in terms of complex hermite polynomials on a technical level we show that finite wh ensembles provide an approximate model for finite polyanalytic ginibre ensembles and we quantify the corresponding deviation by means of this asymptotic description we derive estimates for the rate of convergence of the one point intensity of polyanalytic ginibre ensembles in the thermodynamic limit

universality of general beta ensembles we prove the universality of the ensembles with convex analytic potentials and for any 0 i e we show that the spacing distributions of log gases at any inverse temperature coincide with those of the gaussian ensembles

recognizing solid objects by alignment with an image in this paper we consider the problem of recognizing solid objects from a single two dimensional image of a three dimensional scene we develop a new method for computing a transformation from a three dimensional model coordinate frame to the two dimensional image coordinate frame using three pairs of model and image points we show that this transformation always exists for three noncollinear points and is unique up to a reflective ambiguity the solution method is closed form and only involves second order equations we have implemented a recognition system that uses this transformation method to determine possible alignments of a model with an image each of these hypothesized matches is verified by comparing the entire edge contours of the aligned object with the image edges using the entire edge contours for verification rather than a few local feature points reduces the chance of finding false matches the system has been tested on partly occluded objects in highly cluttered scenes

on universality for orthogonal ensembles of random matrices we prove universality of local eigenvalue statistics in the bulk of the spectrum for orthogonal invariant matrix models with real analytic potentials with one interval limiting spectrum our starting point is the tracy widom formula for the matrix reproducing kernel the key idea of the proof is to represent the differentiation operator matrix written in the basis of orthogonal polynomials as a product of a positive toeplitz matrix and a two diagonal skew symmetric toeplitz matrix

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

on degeneracy in linear complementarity problems abstract let m be an n n matrix and q an n th order vector then the linear complementarity problem lcp q m is defined as follows determine x 0 such that w mx q 0 and x t w 0 a vector x which satisfies these conditions is called a solution of the problem and a solution for which x i w i 0 for at least one value of i is termed degenerate if the solutions of lcp q m are nondegenerate and their number is odd even we say that the solution set has odd even parity and murty has shown that this parity is determined uniquely by m in this paper the idea of parity is extended to degenerate solutions and through these to solution sets containing both degenerate and nondegenerate solutions these results are then used to give a generalization of lemke s method and to analyse the stability of certain degenerate solutions of linear complementarity problems

labeled point pattern matching by fuzzy relaxation abstract a method of matching labeled point patterns is described on the basis of a fuzzy relaxation the method is applicable to labeled point patterns which differ by geometrical transformations such as translation rotation and scale change in this method the point pairs are considered to be the pattern primitives the pattern primitives are geometrically transformed in the process of relaxation so as to minimize a measure of mismatch between the primitive pairs the compatible primitive pairs between the two labeled point patterns are established after only a few iterations of relaxation and the corresponding points are obtained from the primitive pairs as an example the method is applied to the matching of constellations

long time asymptotics of the nonlinear schrodinger equation shock problem the long time asymptotics of two colliding plane waves governed by the focusing nonlinear schrodinger equation are analyzed via the inverse scattering method we find three asymptotic regions in space time a region with the original wave modified by a phase perturbation a residual region with a one phase wave and an intermediate transition region with a modulated two phase wave the leading order terms for the three regions are computed with error estimates using the steepest descent method for riemann hilbert problems the nondecaying initial data requires a new adaptation of this method a new breaking mechanism involving a complex conjugate pair of branch points emerging from the real axis is observed between the residual and transition regions also the effect of the collision is felt in the plane wave state well beyond the shock front at large times 2007 wiley periodicals inc

eidetic wolf search algorithm with a global memory structure abstract a recently proposed metaheuristics called wolf search algorithm wsa has demonstrated its efficacy for various hard to solve optimization problems in this paper an improved version of wsa namely eidetic wsa with a global memory structure gms or just ewsa is presented ewsa makes use of gms for improving its search for the optimal fitness value by preventing mediocre visited places in the search space to be visited again in future iterations inherited from swarm intelligence search agents in ewsa and the traditional wsa merge into an optimal solution although the agents behave and make decisions autonomously heuristic information gathered from collective memory of the swarm search agents is stored in gms the heuristics eventually leads to faster convergence and improved optimal fitness the concept is similar to a hybrid metaheuristics based on wsa and tabu search ewsa is tested with seven standard optimization functions rigorously in particular ewsa is compared with two state of the art metaheuristics ant colony optimization aco and particle swarm optimization pso ewsa shares some similarity with both approaches with respect to directed random search the similarity with aco is however stronger as aco uses pheromones as global information references that allow a balance between using previous knowledge and exploring new solutions under comparable experimental settings identical population size and number of generations ewsa is shown to outperform both aco and pso with statistical significance when dedicating the same computation time only aco can be outperformed due to a comparably long run time per iteration of ewsa

a riemann hilbert problem for skew orthogonal polynomials we find a local d 1 x d 1 riemann hilbert problem characterizing the skew orthogonal polynomials associated to the partition function of orthogonal ensembles of random matrices with a potential function of degree d our riemann hilbert problem is similar to a local dxd riemann hilbert problem found by kuijlaars and mclaughlin characterizing the bi orthogonal polynomials this gives more motivation for finding methods to compute asymptotics of high order riemann hilbert problems and brings us closer to finding full asymptotic expansions of the skew orthogonal polynomials

higher order analogues of the tracy widom distribution and the painleve ii hierarchy we study fredholm determinants related to a family of kernels which describe the edge eigenvalue behavior in unitary random matrix models with critical edge points the kernels are natural higher order analogues of the airy kernel and

limit correlation functions for fixed trace random matrix ensembles universal limits for the eigenvalue correlation functions in the bulk of the spectrum are shown for a class of nondeterminantal random matrices known as the fixed trace or the hilbert schmidt ensemble these universal limits have been proved before for determinantal hermitian matrix ensembles and for some special classes of the wigner random matrices

point matching using asymmetric neural networks abstract matching control points is an important step in many pattern recognition applications the matching problem is formulated under translation and rotation as a 0 1 integer programming problem and an artificial neural network is proposed for approximately solving it the solution to the 0 1 integer programming problem is obtained as the high gain limit point of the continuous network the network is capable of handling distortion and noise and can use both interpoint distance information and feature properties associated with the points the results obtained by the network compare favourably with that of the relaxation method of ton and jain ieee trans geosci remote sensing 27 642 651 1989

the law of large numbers for the maximum of almost gaussian log correlated fields coming from random matrices we compute the leading asymptotics as n rightarrow infty of the maximum of the field q n q log q a n q in mathbb c for any unitarily invariant hermitian random matrix a n associated to a non critical real analytic potential hence we verify the leading order in a conjecture of fyodorov and simm nonlinearity 29 2837 2016 arxiv 1503 07110 math ph formulated for the gue the method relies on a classical upper bound and a more sophisticated lower bound based on a variant of the second moment method which exploits the hyperbolic branching structure of the field q n q q in mathbb h specifically we compare q n to an idealized gaussian field by means of exponential moments in principle this method could also be applied to random fields coming from other point processes provided that one can compute certain mixed exponential moments for unitarily invariant ensembles we show that these assumptions follow from the fyodorov strahov formula fyodorov and strahov in j phys a 36 12 3203 3213 2003 https doi org 10 1088 0305 4470 36 12 320 and asymptotics of orthogonal polynomials derived in deift et al commun pure appl math 52 11 1335 1425 1999 https doi org 10 1002 sici 1097 0312 199911 52 11 3 0 co 2 1