a connection formula for the second painleve transcendent we consider a particular case of the second painleve transcendent y xy 2y 3 it is known that if y x kai x as x then if 0 k 1 y x sim d left x right raise0 5ex hbox scriptstyle 1 kern 0 1em kern 0 15em lower0 25ex hbox scriptstyle 4 sin frac 2 3 left x right 3 2 frac 3 4 d2 ell n left x right c asx to infty where d k and c k are the connection formulae for this nonlinear ordinary differential equation

weighted operators based on dissimilarity function this paper talks about weighted operators based on dissimilarity function and discusses the monotone non decreasingness of these operators i e it looks for conditions leading to aggregation operators moreover the paper discusses the shift invariance and weak monotone non decreasingness of mentioned operators it discusses minimization based operators a w d a g d where w is a weighting vector g is a vector of weighting functions and d is a dissimilarity function d x y f x f y 2 following a w d we recognize the class of arithmetic means ordered weighted averaging owa operators and their extensions operators a w d are monotone non decreasing and hence shift invariant and weak monotone non decreasing too by the operators a g d we introduce a generalization of the operators a w d the operators a g d cover the class of mixture operators quasi mixture operators and their extensions in general these operators need not be non decreasing nor shift invariant and hence nor weak monotone non decreasing

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

on the weak monotonicity of gini means and other mixture functions weak monotonicity was recently proposed as a relaxation of the monotonicity condition for averaging aggregation and weakly monotone functions were shown to have desirable properties when averaging data corrupted with outliers or noise we extended the study of weakly monotone averages by analyzing their transforms and we established weak monotonicity of several classes of averaging functions in particular gini means and mixture operators mixture operators with gaussian weighting functions were shown to be weakly monotone for a broad range of their parameters this study assists in identifying averaging functions suitable for data analysis and image processing tasks in the presence of outliers

vicious walkers and hook young tableaux we consider a generalization of the vicious walker model using a bijection map between the path configuration of the non intersecting random walkers and the hook young diagram we compute the probability concerning the number of movements of the walker applying the saddle point method we reveal that the scaling limit gives the tracy widom distribution which is the same with the limit distribution of the largest eigenvalues of the gaussian unitary ensemble

mixture functions and their monotonicity abstract we consider mixture functions which are a type of weighted averages for which the corresponding weights are calculated by means of appropriate continuous functions of their inputs in general these mixture function need not be monotone increasing for this reason we study sufficient conditions to ensure standard weak and directional monotonicity for specific types of weighting functions we also analyze directional monotonicity when differentiability is assumed

local tail bounds for functions of independent random variables it is shown that functions defined on 0 1 r 1 n satisfying certain conditions of bounded differences that guarantee sub gaussian tail behavior also satisfy a much stronger local sub gaussian property for self bounding and configuration functions we derive analogous locally subexponential behavior the key tool is talagrand s ann probab 22 1994 1576 1587 variance inequality for functions defined on the binary hypercube which we extend to functions of uniformly distributed random variables defined on 0 1 r 1 n for r 2

products of independent quaternion ginibre matrices and their correlation functions we discuss the product of independent induced quaternion 4 ginibre matrices and the eigenvalue correlations of this product matrix the joint probability density function for the eigenvalues of the product matrix is shown to be identical to that of a single ginibre matrix but with a more complicated weight function we find the skew orthogonal polynomials corresponding to the weight function of the product matrix and use the method of skew orthogonal polynomials to compute the eigenvalue correlation functions for product matrices of finite size the radial behavior of the density of states is studied in the limit of large matrices and the macroscopic density is discussed the microscopic limit at the origin at the edge s and in the bulk is also discussed for the radial behavior of the density of states

induced weighted operators based on dissimilarity functions based on the minimization considering dissimilarity function d x y f x f y 2 induced ordered weighted averaging operators iowa and induced ordered generalized mixture operators iom g are discussed in general these operators need not be monotone non decreasing thus they need not be the aggregation operators this paper introduces conditions for their monotonicity and weak monotonicity several examples are included

correlation functions cluster functions and spacing distributions for random matrices the usual formulas for the correlation functions in orthogonal and symplectic matrix models express them as quaternion determinants from this representation one can deduce formulas for spacing probabilities in terms of fredholm determinants of matrix valued kernels the derivations of the various formulas are somewhat involved in this article we present a direct approach which leads immediately to scalar kernels for the unitary ensembles and matrix kernels for the orthogonal and symplectic ensembles and the representations of the correlation functions cluster functions and spacing distributions in terms of them

recursion relations for unitary integrals combinatorics and the toeplitz lattice in a discussion in spring 2001 alexei borodin showed us recursion relations for the toeplitz determinants going with the symbols e t z z 1 and 1 xiz alpha 1 xiz 1 beta borodin obtained these relations using riemann hilbert methods see the recent work of borodin b and baik baik the nature of borodin s recursion relations pointed towards the toeplitz lattice and its virasoro algebra introduced by us in 3 in this paper we take the toeplitz lattice and virasoro algebra approach for a fairly large class of symbols leading to a systematic way of generating recursion relations the latter are very naturally expressed in terms of the l matrices appearing in the toeplitz lattice equations as a surprise we find compared to borodin s a different set of relations except for the 3 step relations associated with the symbol e t z z 1 the painleve analysis of the toeplitz lattice enables us to show the singularity confinement for these recursion relations

a rate of convergence result for the largest eigenvalue of complex white wishart matrices it has been recently shown that if x is an n n matrix whose entries are i i d standard complex gaussian and l1 is the largest eigenvalue of x x there exist sequences mn n and sn n such that l1 mn n sn n converges in distribution to w2 the tracy widom law appearing in the study of the gaussian unitary ensemble this probability law has a density which is known and computable the cumulative distribution function of w2 is denoted f2 in this paper we show that under the assumption that n n 0 we can find a function m continuous and nonincreasing and sequences n n and n n such that for all real s0 there exists an integer n s0 for which if n n n s0 we have with ln n l1 n n n n s s0 n n 2 3 p ln n s f2 s m s0 exp s the surprisingly good 2 3 rate and qualitative properties of the bounding function help explain the fact that the limiting distribution w2 is a good approximation to the empirical distribution of ln n in simulations an important fact from the point of view of e g statistical applications

a limit theorem for shifted schur measures to each partition 1 2 with distinct parts we assign the probability q x p y z where q and p are the schur q functions and z is a normalization constant this measure which we call the shifted schur measure is analogous to the much studied schur measure for the specialization of the first m coordinates of x and the first n coordinates of y equal to 0 1 and the rest equal to zero we derive a limit law for 1 as m n with m n fixed for the schur measure the specialization limit law was derived by johansson our main result implies that the two limit laws are identical

determinantal processes with number variance saturation consider dyson s hermitian brownian motion model after a finite time s where the process is started at n equidistant points on the real line these n points after time s form a determinantal process and has a limit as n this limting determinantal process has the interesting feature that it shows number variance saturation the variance of the number of particles in an interval converges to a limiting value as the length of the interval goes to infinity number variance saturation is also seen for example in the zeros of the riemann function 21 2 the process can also be constructed using non intersecting paths and we consider several variants of this construction one construction leads to a model which shows a transition from a non universal behaviour with number variance saturation to a universal sine kernel behaviour as we go up the line

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

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

affine shuffles shuffles with cuts the whitehouse module and patience sorting abstract type a affine shuffles are compared with riffle shuffles followed by a cut although these probability measures on the symmetric group s n are different they both satisfy a convolution property strong evidence is given that when the underlying parameter q satisfies gcd n q 1 1 the induced measures on conjugacy classes of the symmetric group coincide this gives rise to interesting combinatorics concerning the modular equidistribution by major index of permutations in a given conjugacy class and with a given number of cyclic descents using representation theoretic work on the whitehouse module a formula is obtained for the cycle structure of a riffle shuffle followed by a cut it is proved that the use of cuts does not speed up the convergence rate of riffle shuffles to randomness generating functions for the first pile size in patience sorting from decks with repeated values are derived this relates to random matrices

application of the τ function theory of painleve equations to random matrices piv pii and the gue tracy and widom have evaluated the cumulative distribution of the largest eigenvalue for the finite and scaled infinite gue in terms of a piv and pii transcendent respectively we generalise these results to the evaluation of where for and otherwise and the average is with respect to the joint eigenvalue distribution of the gue as well as to the evaluation of of particular interest are and f n 2 and their scaled limits which give the distribution of the largest eigenvalue and the density respectively our results are obtained by applying the okamoto function theory of piv and pii for which we give a self contained presentation based on the recent work of noumi and yamada we point out that the same approach can be used to study the quantities and f n a for the other classical matrix ensembles

weighted aggregation operators based on minimization minimization based aggregation operators a g d are discussed special attention is paid to weighting function g based cases related to some fixed dissimilarity function d when d 2 x y x y 2 we recognize mixture operators and we recall some sufficient conditions for g ensuring the monotonicity of a g d for d 1 x y x y and non decreasing non increasing g a g d is shown to be the upper lower median whenever a g d is an aggregation operator

shape fluctuations and random matrices we study a certain random growth model in two dimensions closely related to the one dimensional totally asymmetric exclusion process the results show that the shape fluctuations appropriately scaled converges in distribution to the tracy widom largest eigenvalue distribution for the gaussian unitary ensemble gue

on the distribution of the length of the longest increasing subsequence of random permutations let sn be the group of permutations of 1 2 n if 7r e sn we say that 7 i1 7f ik is an increasing subsequence in 7r if il i2 ik and 7r ii 7r i2 7r ik let 1n r be the length of the longest increasing subsequence for example if n 5 and 7r is the permutation 5 1 3 2 4 in one line notation thus 7r 1 5 7r 2 1 then the longest increasing subsequences are 1 2 4 and 1 3 4 and n 3 equip sn with uniform distribution

a path transformation for random walks and the robinson schensted correspondence the author and marc yor recently introduced a path transformation g k with the property that for x belonging to a certain class of random walks on z k the transformed walk g k x has the same law as the original walk conditioned never to exit the weyl chamber x x 1 less than or equal to less than or equal to x k in this paper we show that g k is closely related to the robinson schensted algorithm and use this connection to give a new proof of the above representation theorem the new proof is valid for a larger class of random walks and yields additional information about the joint law of x and g k x the corresponding results for the brownian model are recovered by donsker s theorem these are connected with hermitian brownian motion and the gaussian unitary ensemble of random matrix theory the connection we make between the path transformation g k and the robinson schensted algorithm also provides a new formula and interpretation for the latter this can be used to study properties of the robinson schensted algorithm and moreover extends easily to a continuous setting

how instanton combinatorics solves painleve vi v and iiis we elaborate on a recently conjectured relation of painlev transcendents and 2d conformal field theory general solutions of painlev vi v and iii are expressed in terms of c 1 conformal blocks and their irregular limits agt related to instanton partition functions in supersymmetric gauge theories with nf 0 1 2 3 4 the resulting combinatorial series representations of painlev functions provide an efficient tool for their numerical computation at finite values of the argument the series involves sums over bipartitions which in the simplest cases coincide with gessel expansions of certain toeplitz determinants considered applications include fredholm determinants of classical integrable kernels scaled gap probability in the bulk of the gaussian unitary ensemble and all order conformal perturbation theory expansions of correlation functions in the sine gordon field theory at the free fermion point

conditioned random walks and the rsk correspondence we consider the stochastic evolution of three variants of the rsk algorithm giving both analytic descriptions and probabilistic interpretations symmetric functions play a key role and the probabilistic interpretations are obtained by elementary doob hunt theory in each case the evolution of the shape of the tableau obtained via the rsk algorithm can be interpreted as a conditioned random walk this is intuitively appealing and can be used for example to obtain certain relationships between orthogonal polynomial ensembles in a certain scaling limit there is a continous version of the rsk algorithm which inherits much of the structure exhibited in the discrete settings intertwining relationships between conditioned and unconditioned random walks are also given in the continuous limit these are related to the harish chandra itzyksen zuber integral

a growth model in a random environment we consider a model of interface growth in two dimensions given by a height function on the sites of the one dimensional integer lattice according to the discrete time update rule the height above the site x increases to the height above x 1 if the latter height is larger otherwise the height above x increases by 1 with probability p x we assume that p x are chosen independently at random with a common distribution f and that the initial state is such that the origin is far above the other sites we explicitly identify the asymptotic shape and prove that in the pure regime the fluctuations about that shape normalized by the square root of time are asymptotically normal this contrasts with the quenched version conditioned on the environment and normalized by the cube root of time the fluctuations almost surely approach a distribution known from random matrix theory

limit theorems for height fluctuations in a class of discrete space and time growth models we introduce a class of one dimensional discrete space discrete time stochastic growth models described by a height function ht x with corner initialization we prove with one exception that the limiting distribution function of ht x suitably centered and normalized equals a fredholm determinant previously encountered in random matrix theory in particular in the universal regime of large x and large t the limiting distribution is the fredholm determinant with airy kernel in the exceptional case called the critical regime the limiting distribution seems not to have previously occurred the proofs use the dual rsk algorithm gessel s theorem the borodin okounkov identity and a novel rigorous saddle point analysis in the fixed x large t regime we find a brownian motion representation this model is equilvalent to the seppalainen johansson model hence some of our results are not new but the proofs are

integrable fredholm operators and dual isomonodromic deformations the fredholm determinants of a special class of integrable integral operators k supported on the union of m curve segments in the complex plane are shown to be the functions of an isomonodromic family of meromorphic covariant derivative operators having regular singular points at the 2m endpoints of the curve segments and a singular point of poincare index 1 at infinity the rank r of the corresponding vector bundle over the riemann sphere equals the number of distinct terms in the exponential sum defining the numerator of the integral kernel the matrix riemann hilbert problem method is used to deduce an identification of the fredholm determinant as a function in the sense of segal wilson and sato i e in terms of abelian group actions on the determinant line bundle over a loop space grassmannian an associated dual isomonodromic family of covariant derivative operators having rank n 2m and r finite regular singular points located at the values of the exponents defining the kernel of k is derived the deformation equations for this family are shown to follow from an associated dual set of riemann hilbert data in which the roles of the r exponential factors in the kernel and the 2m endpoints of its support are interchanged the operators are analogously associated to an integral operator whose fredholm determinant is equal to that of k

janossy densities of coupled random matrices we explicitly calculate the janossy densities for a special class of finite determinantal random point processes with several types of particles introduced by prahofer and spohn and in the full generality by johansson in connection with the analysis of polynuclear growth models the results of this paper generalize the theorem we proved earlier with borodin about the janossy densities in biorthogonal ensembles in particular our results can be applied to ensembles of random matrices coupled in a chain which provide a very important example of determinantal ensembles we study

construction of image reduction operators using averaging aggregation functions in this work we present an image reduction algorithm based on averaging aggregation functions we axiomatically define the concepts of image reduction operator and local reduction operator we study the construction of the latter by means of averaging functions and we propose an image reduction algorithm image reduction operator we analyze the properties of several averaging functions and their effect on the image reduction algorithm finally we present experimental results where we apply our algorithm in two different applications analyzing the best operators for each concrete application

weak monotonicity of lehmer and gini means we analyze directional monotonicity of several mixture functions in the direction 1 1 1 called weak monotonicity our particular focus is on power weighting functions and the special cases of lehmer and gini means we establish limits on the number of arguments of these means for which they are weakly monotone these bounds significantly improve the earlier results and hence increase the range of applicability of gini and lehmer means we also discuss the case of affine weighting functions and find the smallest constant which ensures weak monotonicity of such mixture functions

edge statistics for a class of repulsive particle systems we study a class of interacting particle systems on mathbb r which was recently investigated by gotze and venker ann probab 42 6 2207 2242 2014 doi 10 1214 13 aop844 these ensembles generalize eigenvalue ensembles of hermitian random matrices by allowing different interactions between particles although these ensembles are not known to be determinantal one can use the stochastic linearization method of gotze and venker ann probab 42 6 2207 2242 2014 doi 10 1214 13 aop844 to represent them as averages of determinantal ones our results describe the transition between universal behavior in the regime of the tracy widom law and non universal behavior for large deviations of the rightmost particle moreover a detailed analysis of the transition that occurs in the regime of moderate deviations is provided we also compare our results with the corresponding ones obtained recently for determinantal ensembles eichelsbacher et al in symmetry integr geom methods appl 12 18 2016 doi 10 3842 sigma 2016 093 schuler in moderate large and superlarge deviations for extremal eigenvalues of unitarily invariant ensembles ph d thesis university of bayreuth 2015 in particular we discuss how the averaging effects the leading order behavior in the regime of large deviations in the analysis of the averaging procedure we use detailed asymptotic information on the behavior of christoffel darboux kernels that is uniform for perturbative families of weights such results have been provided by k schubert k schuler and the authors in kriecherbauer et al markov process relat fields 21 3 part 2 639 694 2015

application of the τ function theory of painleve equations to random matrices pv piii the lue jue and cue with denoting an average with respect to the eigenvalue pdf for the laguerre unitary ensemble the object of our study is e n i a x l i l 0 i l for i 0 s and i s where x i l 1 for l i and x i l 0 other wise using okamoto s development of the theory of the painleve v equation it is shown that e n i a is a function associated with the hamiltonian therein and so can be characterized as the solution of a certain second order second degree differential equation or in terms of the solution of certain difference equations the cases 0 and 2 are of particular interest because they correspond to the cumulative distribution and density function respectively for the smallest and largest eigenvalue in the case i s e n i a is simply related to an average in the jacobi unitary ensemble and this in turn is simply related to certain averages over the orthogonal group the unitary symplectic group and the circular unitary ensemble the latter integrals are of interest for their combinatorial content also considered are the hard edge and soft edge scaled limits of e n i a in particular in the hard edge scaled limit it is shown that the limiting quantity e hard 0 s a can be evaluated as a function associated with the hamiltonian in okamoto s theory of the painleve iii equation

distribution of the shape of markovian random words the distribution of the shape of the semi standard tableau of a random word in k letters is asymptotically given by the distribution of the spectrum of a random traceless k k gaussian unitary ensemble gue matrix provided that these letters are independent with uniform distribution kuperberg 2002 conjectured that this result by johansson 2001 remains valid if the letters of the word are generated by an irreducible markov chain on the alphabet with cyclic transition matrix in this paper we give a proof of this conjecture for an alphabet with k 2 letters

parametric conditions for a monotone tsk fuzzy inference system to be an n ary aggregation function despite the popularity and practical importance of the fuzzy inference system fis the use of an fis model as an n ary aggregation function which is characterized by both the monotonicity and boundary properties is yet to be established this is because research on ensuring that fis models satisfy the monotonicity property i e monotone fis is relatively new not to mention the additional requirement of satisfying the boundary property the aim of this paper therefore is to establish the parametric conditions for the takagi sugeno kang tsk fis model to operate as an n ary aggregation function hereafter denoted as n tsk fis via the specifications of fuzzy membership functions fmfs and fuzzy rules an absorption property with fuzzy rules interpretation is outlined and the use of n tsk fis as a uni norm is explained exploiting the established parametric conditions a framework for which an n tsk fis model can be constructed from data samples is formulated and analyzed along with a number of remarks synthetic data sets and a benchmark example on education assessment are presented and discussed to be best of the authors knowledge this study serves as the first use of the tsk fis model as an n ary aggregation function

longest increasing subsequences from patience sorting to the baik deift johansson theorem we describe a simple one person card game patience sorting its analysis leads to a broad circle of ideas linking young tableaux with the longest increasing subsequence of a random permutation via the schensted correspondence a recent highlight of this area is the work of baik deift johansson which yields limiting probability laws via hard analysis of toeplitz determinants

a model for the bus system in cuernavaca mexico the bus system in cuernavaca mexico and its connections to random matrix distributions have been the subject of an interesting recent study by m krbalek and p seba in 15 16 in this paper we introduce and analyse a microscopic model for the bus system we show that introducing a natural repulsion does produce random matrix distributions in natural double scaling regimes the techniques employed include non intersecting paths logarithmic potential theory determinantal point processes and asymptotic analysis of several orthogonal polynomial ensembles in addition we introduce a circular bus model and include various calculations of non crossing probabilities

algebraic aspects of increasing subsequences we present a number of results relating partial cauchy littlewood sums integrals over the compact classical groups and increasing subsequences of permutations these include integral formulae for the distribution of the longest increasing subsequence of a random involution with constrained number of fixed points new formulae for partial cauchy littlewood sums as well as new proofs of old formulae relations of these expressions to orthogonal polynomials on the unit circle and explicit bases for invariant spaces of the classical groups together with appropriate generalizations of the straightening algorithm

zeros of the jimbo miwa ueno tau function we introduce a family of local deformations for meromorphic connections on p1 in the neighborhood of a higher rank simple singularity following the scheme in malgrange mathematique et physique progress in mathematics birkhauser boston 1983 vol 37 pp 381 400 ibid pp 401 426 ibid pp 427 438 we use these local models to prove that the zeros of the tau function introduced by jimbo miwa and ueno in their pioneering work on birkhoff deformations at irregular singular points physica d 2 306 352 2 407 448 1981 4 26 46 1983 publ rims kyoto univ 17 2 703 721 1981 occur at precisely those points in the deformation space at which a certain birkhoff riemann hilbert problem fails to have a solution

matrix models and stochastic growth in donaldson thomas theory we show that the partition functions which enumerate donaldson thomas invariants of local toric calabi yau threefolds without compact divisors can be expressed in terms of specializations of the schur measure we also discuss the relevance of the hall littlewood and jack measures in the context of bps state counting and study the partition functions at arbitrary points of the kahler moduli space this rewriting in terms of symmetric functions leads to a unitary one matrix model representation for donaldson thomas theory we describe explicitly how this result is related to the unitary matrix model description of chern simons gauge theory this representation is used to show that the generating functions for donaldson thomas invariants are related to tau functions of the integrable toda and toeplitz lattice hierarchies the matrix model also leads to an interpretation of donaldson thomas theory in terms of non intersecting paths in the lock step model of vicious walkers we further show that these generati

universality at the edge of the spectrum in wigner random matrices we prove universality at the edge for rescaled correlation functions of wigner random matrices in the limit n as a corollary we show that after proper rescaling the 1th 2nd 3rd etc eigenvalues of wigner random hermitian resp real symmetric matrix weakly converge to the distributions established by tracy and widom in g u e g o e cases

on the distributions of the lengths of the longest monotone subsequences in random words we consider the distributions of the lengths of the longest weakly increasing and strongly decreasing subsequences in words of length n from an alphabet of k letters in the limit as k these become the corresponding distributions for permutations on n letters we find toeplitz determinant representations for the exponential generating functions on n of these distribution functions and show that they are expressible in terms of solutions of painleve v equations we show further that in the weakly increasing case the generating unction gives the distribution of the smallest eigenvalue in the k k laguerre random matrix ensemble and that the distribution itself has after centering and normalizing an n limit which is equal to the distribution function for the largest eigenvalue in the gaussian unitary ensemble of k k hermitian matrices of trace zero

generalized christoffel darboux formula for skew orthogonal polynomials and random matrix theory we obtain a generalized christoffel darboux gcd formula for skew orthogonal polynomials using this we present an alternative derivation of the level density and two point function for gaussian orthogonal ensembles and gaussian symplectic ensembles of random matrices

multivariate analysis and jacobi ensembles largest eigenvalue tracy widom limits and rates of convergence abstract let a and b be independent central wishart matrices in p variables with common covariance and having m and n degrees of freedom respectively the distribution of the largest eigenvalue of a b 1 b has numerous applications in multivariate statistics but is difficult to calculate exactly suppose that m and n grow in proportion to p we show that after centering and scaling the distribution is approximated to second order o p 2 3 by the tracy widom law the results are obtained for both complex and then real valued data by using methods of random matrix theory to study the largest eigenvalue of the jacobi unitary and orthogonal ensembles asymptotic approximations of jacobi polynomials near the largest zero play a central role

stability of interval matrix polynomial in continuous and discrete cases abstract sufficient conditions of asymptotic stability of interval matrix polynomial of both continuous and discrete cases are given these conditions are presented in terms of the positive definiteness of either an ln ln or l l hermitian and interval matrix therefore the computation of the determinant interval polynomial is avoided

random words toeplitz determinants and integrable systems ii this paper connects the analysis of the length of the longest weakly increasing subsequence of inhomogeneous random words to a riemnn hilbert problem and an associated system of integrable partial differential equations in particular we show that the poissonization of the distribution function of this length can be identified as the jimbo miwa ueno tau function

discrete orthogonal polynomial ensembles and the plancherel measure we consider discrete orthogonal polynomial ensembles which are discrete analogues of the orthogonal polynomial ensembles in random matrix theory these ensembles occur in certain problems in combinatorial probability and can be thought of as probability measures on partitions the meixner ensemble is related to a two dimensional directed growth model and the charlier ensemble is related to the lengths of weakly increasing subsequences in random words the krawtchouk ensemble occurs in connection with zig zag paths in random domino tilings of the aztec diamond and also in a certain simplifled directed flrst passage percolation model we use the charlier ensemble to investigate the asymptotics of weakly increasing subsequences in random words and to prove a conjecture of tracy and widom as a limit of the meixner ensemble or the charlier ensemble we obtain the plancherel measure on partitions and using this we prove a conjecture of baik deift and johansson that under the plancherel measure the distribution of the lengths of the flrst k rows in the partition appropriately scaled converges to the asymptotic joint distribution for the k largest eigenvalues of a random matrix from the gaussian unitary ensemble in this problem a certain discrete kernel which we call the discrete bessel kernel plays an important role

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

the partition function of the two matrix model as an isomonodromic τ function we consider the itzykson zuber eynard mehta two matrix model and prove that the partition function is an isomonodromic function in a sense that generalizes that of jimbo et al monodromy preserving deformation of linear ordinary differential equations with rational coefficients physica d 2 306 1981 in order to achieve the generalization we need to define a notion of function for isomonodromic systems where the adregularity of the leading coefficient is not a necessary requirement

generalized mixture operators using weighting functions a comparative study with wa and owa abstract in the context of multiple attribute decision making we present an aggregation scheme based on generalized mixture operators using weighting functions and we compare it with two standard aggregation methods weighted averaging and ordered weighted averaging specifically we consider linear and quadratic weight generating functions that penalize bad attribute performances and reward good attribute performances an illustrative example borrowed from the literature is used to perform the operators comparison we believe that this comparative study will highlight the potential and flexibility of generalized mixture operators using weighting functions that depend on attribute performances

asymptotic values for degrees associated with strips of young diagrams abstract the sums s l n occur in the representations of the symmetric and the general linear groups in combinatorics and in pi algebras we give asymptotic values for these sums

a general theory for matrix root clustering in subregions of the complex plane we consider the general problem of root clustering of a matrix in the complex plane let a in c n times n and s subset c find the largest class of s and an algebraic criterion which is necessary and sufficient for lambda i a in s i 1 2 n we introduce two types of regions which constitute the largest class of s known to date the criterion is presented both for open regions and closed ones the results are used to define a design methodology for control systems moreover all classical results are shown to be special cases of the present theory

the ginibre ensemble of real random matrices and its scaling limits we give a closed form for the correlation functions of ensembles of a class of asymmetric real matrices in terms of the pfaffian of an antisymmetric matrix formed from a 2 2 matrix kernel associated to the ensemble we apply this result to the real ginibre ensemble and compute the bulk and edge scaling limits of its correlation functions as the size of the matrices becomes large

a fredholm determinant identity and the convergence of moments for random young tableaux we obtain an identity between fredholm determinants of two kinds of operators one acting on functions on the unit circle and the other acting on functions on a subset of the integers this identity is a generalization of an identity between a toeplitz determinant and a fredholm determinant that has appeared in the random permutation context using this identity we prove in particular convergence of moments for arbitrary rows of a random young diagram under plancherel measure

displacement convexity for the generalized orthogonal ensemble the generalized orthogonal ensemble of n n real symmetric matrices x has probability measure nu n left dx right z n 1 exp left ntracev left x right right where dx is the product of lebesgue measure on the matrix entries and v left x right geqslant left 2 delta right log left x right with 0 the eigenvalue distribution is concentrated on left a mathord left vphantom a 2 right kern nulldelimiterspace 2 a mathord left vphantom a 2 right kern nulldelimiterspace 2 right for some a 0 or if the variance of the trace is o 1 n2 then the empirical distribution of eigenvalues converges weakly almost surely to some non random probability measure on a 2 a 2 as o left sqrt log n mathord left vphantom n n 2 right kern nulldelimiterspace n 2 right these conditions are satisfied for certain polynomial potentials the logarithmic energy is displacement convex as a functional on charge distributions with fixed mean along the real line when the trace distribution satisfies a logarithmic sobolev inequality or equivalently a quadratic transportation inequality the joint eigenvalue distributions and the limiting equilibrium measure likewise satisfy quadratic transportation inequalities in the sense of talagrand 24

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

stability of a matrix polynomial in continuous systems two sufficient conditions under which the roots of the determinant of a given m times m matrix polynomial of n th order lie in the open left half plane have been obtained the first condition is given in terms of the positive definiteness of an mn times mn symmetric matrix while the second condition is given in terms of the positive definiteness of an m times m matrix that is a function of s re s leq 0 these conditions are represented in terms of rational functions of the coefficient matrices of the given matrix polynomial therefore the explicit computation of the determinant polynomial is not required

integrable structure of ginibre s ensemble of real random matrices and a pfaffian integration theorem in the recent publication e kanzieper and g akemann in phys rev lett 95 230201 2005 an exact solution was reported for the probability pn k to find exactly k real eigenvalues in the spectrum of an n n real asymmetric matrix drawn at random from ginibre s orthogonal ensemble ginoe in the present paper we offer a detailed derivation of the above result by concentrating on the proof of the pfaffian integration theorem the key ingredient of our analysis of the statistics of real eigenvalues in the ginoe we also initiate a study of the correlations of complex eigenvalues and derive a formula for the joint probability density function of all complex eigenvalues of a ginoe matrix restricted to have exactly k real eigenvalues in the particular case of k 0 all correlation functions of complex eigenvalues are determined

hammersley s interacting particle process and longest increasing subsequences in a famous paper 8 hammersley investigated the lengthl n of the longest increasing subsequence of a randomn permutation implicit in that paper is a certain one dimensional continuous space interacting particle process by studying a hydrodynamical limit for hammersley s process we show by fairly soft arguments that limn 1 2 el n 2 this is a known result but previous proofs 14 11 relied on hard analysis of combinatorial asymptotics

stability of a matrix polynomial in discrete systems two sufficient conditions that the determinant of a nonsingular real m times m matrix polynomial of n th order has all its roots inside the unit circle have been obtained these conditions are represented in terms of rational functions of the coefficient matrices therefore these conditions do not require the computation of the determinant polynomial the first condition is given in terms of the positive definiteness of an mn times mn symmetric matrix while the second condition is expressed by the positive definiteness of an m times m hermitian matrix which is a function of z z leq 1 the first condition implies the second and hence is more restrictive than the second

monodromy preserving deformation of linear ordinary differential equations with rational coefficients iii abstract a unified treatment of monodromy and spectrum preserving deformations is presented in particular a general procedure is described to reduce the latter into the former consistently the concept of the function previously introduced for the former 2 is extended to the isospectral context it is shown that the general monodromy and spectrum preserving deformation equations can be written as hirota s bilinear differential equations by using the functions as dependent variables

differential equations for dyson processes we call a dyson process any process on ensembles of matrices in which the entries undergo diffusion we are interested in the distribution of the eigenvalues or singular values of such matrices in the original dyson process it was the ensemble of n n hermitian matrices and the eigenvalues describe n curves given sets x1 xm the probability that for each k no curve passes through xk at time k is given by the fredholm determinant of a certain matrix kernel the extended hermite kernel for this reason we call this dyson process the hermite process similarly when the entries of a complex matrix undergo diffusion we call the evolution of its singular values the laguerre process for which there is a corresponding extended laguerre kernel scaling the hermite process at the edge leads to the airy process which was introduced by prahofer and spohn as the limiting stationary process for a polynuclear growth model and in the bulk to the sine process scaling the laguerre process at the edge leads to the bessel process

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

aggregation with generalized mixture operators using weighting functions this paper regards weighted aggregation operators in multiple attribute decision making and its main goal is to investigate ways in which weights can depend on the satisfaction degrees of the various attributes criteria we propose and discuss two types of weighting functions that penalize poorly satisfied attributes and reward well satisfied attributes we discuss in detail the characteristics and properties of both functions moreover we present an illustrative example to clarify the use and behaviour of such weighting functions comparing the results with those of standard weighted averaging operators

emergence of a singularity for toeplitz determinants and painleve v we obtain asymptotic expansions for toeplitz determinants corresponding to a family of symbols depending on a parameter t for t positive the symbols are regular so that the determinants obey szeg s strong limit theorem if t 0 the symbol possesses a fisher hartwig singularity letting t 0 we analyze the emergence of a fisher hartwig singularity and a transition between the two different types of asymptotic behavior for toeplitz determinants this transition is described by a special painleve v transcendent a particular case of our result complements the classical description of wu mccoy tracy and barouch of the behavior of a 2 spin correlation function for a large distance between spins in the two dimensional ising model as the phase transition occurs

exact results and universal asymptotics in the laguerre random matrix ensemble for various values of the parameters in the laguerre random matrix ensemble the distribution of the smallest eigenvalue and the scaled n level distribution function are calculated exactly in terms of generalized hypergeometric functions in certain cases these functions are expressed as multidimensional integrals from which asymptotic formulas are calculated and predictions of universal behavior verified

asymptotics of plancherel measures for symmetric groups 1 1 plancherel measures given a finite group g by the corresponding plancherel measure we mean the probability measure on the set g of irreducible representations of g which assigns to a representation g the weight dim g for the symmetric group s n the set s n is the set of partitions of the number n which we shall identify with young diagrams with n squares throughout this paper the plancherel measure on partitions arises naturally in representation theoretic combinatorial and probabilistic problems for example the plancherel distribution of the first part of a partition coincides with the distribution of the longest increasing subsequence of a uniformly distributed random permutation 31 we denote the plancherel measure on partitions of n by mn

random unitary matrices permutations and painleve this paper is concerned with certain connections between the ensemble of n n unitary matrices specifically the characteristic function of the random variable tr u and combinatorics specifically ulam s problem concerning the distribution of the length of the longest increasing subsequence in permutation groups and the appearance of painleve functions in the answers to apparently unrelated questions among the results is a representation in terms of a painleve v function for the characteristic function of tr u and using recent results of baik deift and johansson an expression in terms of a painleve ii function for the limiting distribution of the length of the longest increasing subsequence in the hyperoctahedral groups