a pde for the multi time joint probability of the airy process abstract this paper gives a pde for multi time joint probability of the airy process which generalizes adler and van moerbeke s result on the 2 time case as an intermediate step the pde for the multi time joint probability of the dyson brownian motion is also given

level spacing distributions and the bessel kernel scaling models of randomn n hermitian matrices and passing to the limitn leads to integral operators whose fredholm determinants describe the statistics of the spacing of the eigenvalues of hermitian matrices of large order for the gaussian unitary ensemble and for many others as well the kernel one obtains by scaling in the bulk of the spectrum is the sine kernel frac sin pi x y pi x y rescaling the gue at the edge of the spectrum leads to the kernel frac ai x ai y ai x ai y x y where ai is the airy function in previous work we found several analogies between properties of this airy kernel and known properties of the sine kernel a system of partial differential equations associated with the logarithmic differential of the fredholm determinant when the underlying domain is a union of intervals a representation of the fredholm determinant in terms of a painleve transcendent in the case of a single interval and also in this case asymptotic expansions for these determinants and related quantities achieved with the help of a differential operator which commutes with the integral operator in this paper we show that there are completely analogous properties for a class of kernels which arise when one rescales the laguerre or jacobi ensembles at the edge of the spectrum namely frac j alpha sqrt x sqrt y j alpha sqrt y sqrt x j alpha sqrt x j alpha sqrt y 2 x y wherej z is the bessel function of order in the cases 1 2 these become after a variable change the kernels which arise when taking scaling limits in the bulk of the spectrum for the gaussian orthogonal and symplectic ensembles in particular an asymptotic expansion we derive will generalize ones found by dyson for the fredholm determinants of these kernels

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

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

complex wishart matrices and conductance in mesoscopic systems exact results it is noted that the hypothesis of independent random complex elements for the off diagonal blocks say m2 and m3 of the transfer matrix describing conductance in a mesoscopic wire allows the eigenvalue distribution of the matrix product m3 m3 or m2 m2 to be computed exactly exact expressions in terms of double wronskian and toeplitz determinants are derived for the distribution of the smallest and second smallest eigenvalue of this and similar random matrix products

on pfaffian random point fields we study pfaffian random point fields by using the moore dyson quaternion determinants first we give sufficient conditions that ensure that a self dual quaternion kernel defines a valid random point field and then we prove a clt for pfaffian point fields the proofs are based on a new quaternion extension of the cauchy binet determinantal identity in addition we derive the fredholm determinantal formulas for the pfaffian point fields which use the quaternion determinant

steady turbulent flow computations using a low mach fully compressible scheme a recently proposed modification to fully compressible schemes significantly improves the resolution of low mach number features tackling the problem of excessive numerical dissipation as the mach number reduces this paper explores the application of this modification to reynolds averaged navier stokes simulations using second and fifth order in space schemes following verification of the modeling approach four test cases are employed to highlight the scheme performance including a backward facing step a two dimensional lid driven cavity the naca 4412 airfoil incorporating a trailing edge separation and finally a mach 2 25 oblique shock wave boundary layer interaction it demonstrates first that the converged solution is grid and scheme independent as expected and that the low mach number correction provides a significant improvement in low mach number regions because this correction is easy to implement computationally inexpensive and stable the conclusion is that it should be recommend

non intersecting paths random tilings and random matrices we investigate certain measures induced by families of non intersecting paths in domino tilings of the aztec diamond rhombus tilings of an abc hexagon a dimer model on a cylindrical brick lattice and a growth model the measures obtained e g the krawtchouk and hahn ensembles have the same structure as the eigenvalue measures in random matrix theory like gue which can in fact can be obtained from non intersecting brownian motions the derivations of the measures are based on the karlin mcgregor or lindstrom gessel viennot method we use the measures to show some asymptotic results for the models

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

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

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

continuous and discrete painleve equations arising from the gap probability distribution of the finite n gaussian unitary ensembles in this paper we study the gap probability problem in the gaussian unitary ensembles of n by n matrices the probability that the interval j a a is free of eigenvalues in the works of tracy and widom adler and van moerbeke and forrester and witte on this subject it has been shown that two painleve type differential equations arise in this context the first is the jimbo miwa okomoto sigma form and the second is a particular painleve iv using the ladder operator technique of orthogonal polynomials we derive three quantities associated with the gap probability denoted as sigma n a r n a and r n a we show that each one satisfies a second order painleve type differential equation as well as a discrete painleve type equation in particular in addition to providing an elementary derivation of the aforementioned sigma form and painleve iv we are able to show that the quantity r n a satisfies a particular case of chazy s second degree second order differential equation for the discrete equations we show that the quantity r n a satisfies a particular form of the modified discrete painleve ii equation obtained by grammaticos and ramani in the context of backlund transformations we also derive second order second degree difference equations for the quantities r n a and sigma n a

phase transitions and equilibrium measures in random matrix models the paper is devoted to a study of phase transitions in the hermitian random matrix models with a polynomial potential in an alternative equivalent language we study families of equilibrium measures on the real line in a polynomial external field the total mass of the measure is considered as the main parameter which may be interpreted also either as temperature or time our main tools are differentiation formulas with respect to the parameters of the problem and a representation of the equilibrium potential in terms of a hyperelliptic integral using this combination we introduce and investigate a dynamical system system of odes describing the evolution of families of equilibrium measures on this basis we are able to systematically derive a number of new results on phase transitions such as the local behavior of the system at all kinds of phase transitions as well as to review a number of known ones

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

janossy densities ii pfaffian ensembles we extend the main result of the companion paper j stat phys 113 595 610 to the case of the pfaffian ensembles

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

symmetric polynomials generalized jacobi trudi identities and τ functions an element grnh f of the grassmannian of n dimensional subspaces of the hardy space h h2 extended over the field f c x1 xn may be associated to any polynomial basis 0 1 for c x the plucker coordinates s n x1 xn of labeled by partitions provide an analog of jacobi s bi alternant formula defining a generalization of schur polynomials applying the recursion relations satisfied by the polynomial system to the analog hi 0 of the complete symmetric functions generates a doubly infinite matrix hi j of symmetric polynomials that determine an element h grn h f this is shown to coincide with implying a set of generalized jacobi identities extending a result obtained by sergeev and veselov moscow math j 14 161 168 2014 for the case of orthogonal polynomials the symmetric polynomials s n x1 xn are shown to be kp kadomtsev petviashvili functions in terms of the power sums x of the xa s viewed as kp flow variables a fermionic operator representation is derived for these as well as for the infinite sums s n x s n t associated to any pair of polynomial bases which are shown to be 2d toda lattice functions a number of applications are given including classical group character expansions matrix model partition functions and generators for random processes an element grnh f of the grassmannian of n dimensional subspaces of the hardy space h h2 extended over the field f c x1 xn may be associated to any polynomial basis 0 1 for c x the plucker coordinates s n x1 xn of labeled by partitions provide an analog of jacobi s bi alternant formula defining a generalization of schur polynomials applying the recursion relations satisfied by the polynomial system to the analog hi 0 of the complete symmetric functions generates a doubly infinite matrix hi j of symmetric polynomials that determine an element h grn h f this is shown to coincide with implying a set of generalized jacobi identities extending a result obtained by sergeev and veselov moscow math j 14 161 168 2014 for the case of orthogonal polynomials the symmetric polynomials s n x1 xn are shown to be kp kadomtsev petviashvili functions in terms of the power sums x of the xa s viewed as kp flow variables a fermionic operator re

rank 1 real wishart spiked model in this paper we consider n dimensional real wishart matrices y in the class input amssym w bbb r sigma m in which all but one eigenvalue of is 1 let the nontrivial eigenvalue of be 1 then as n m with m n 2 finite and nonzero the eigenvalue distribution of y will converge into the marchenko pastur distribution inside a bulk region when increases from 0 one starts to see a stray eigenvalue of y outside of the support of the marchenko pastur density as this stray eigenvalue leaves the bulk region a phase transition will occur in the largest eigenvalue distribution of the wishart matrix in this paper we will compute the asymptotics of the largest eigenvalue distribution when the phase transition occurs we will first establish the results that are valid for all n and m and will use them to carry out the asymptotic analysis in particular we have derived a contour integral formula for the harish chandra itzykson zuber integral when x and y are real symmetric and y is a rank 1 matrix this allows us to write down a fredholm determinant formula for the largest eigenvalue distribution and analyze it using orthogonal polynomial techniques as a result we obtain an integral formula for the largest eigenvalue distribution in the large n limit characterized by painleve transcendents the approach used in this paper is very different from a recent paper by bloemenal and virag in which the largest eigenvalue distribution was obtained using a stochastic operator method in particular the painleve formula for the largest eigenvalue distribution obtained in this paper is new 2012 wiley periodicals inc

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

janossy densities i determinantal ensembles we derive an elementary formula for janossy densities for determinantal point processes with a finite rank projection type kernel in particular for 2 polynomial ensembles of random matrices we show that the janossy densities on an interval i can be expressed in terms of the christoffel darboux kernel for the orthogonal polynomials on the complement of i

explicit representation of finite predictor coefficients and its applications we consider the finite past predictor coefficients of stationary time series and establish an explicit representation for them in terms of the ma and ar coefficients the proof is based on the alternate applications of projection operators associated with the infinite past and the infinite future applying the result to long memory processes we give the rate of convergence of the finite predictor coefficients and prove an inequality of baxter type

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

kernels and point processes associated with whittaker functions this article considers whittaker s confluent hypergeometric function w where is real and is real or purely imaginary then x x 1 2w x arises as the scattering function of a continuous time linear system with state space l2 1 2 and input and output spaces c the hankel operator on l2 0 is expressed as a matrix with respect to the laguerre basis and gives the hankel matrix of moments of a jacobi weight w0 x xb 1 x a the operation of translating is equivalent to deforming w0 to give wt x e t xxb 1 x a the determinant of the hankel matrix of moments of we satisfies the form of painleve s transcendental differential equation pv it is shown that gives rise to the whittaker kernel from random matrix theory as studied by borodin and olshanski commun math phys 211 335 358 2000 whittaker kernels are closely related to systems of orthogonal polynomials for a pollaczek jacobi type weight lying outside the usual szego class

on semiclassical linear functionals integral representations abstract if is a functional of moments and satisfies the equation d o where x and x are arbitrary polynomials but with the only condition deg 1 then is said to be a semiclassical functional when is semiclassical and regular its corresponding orthogonal polynomial sequence ops pn x n 0 is called a semiclassical ops in this paper integral representations are given for semiclassical functionals regular or not in cases a deg deg and b deg deg but in the latter case when deg 0 the fundamental result is the following if n with n 0 are the moments of an a functional then n cm n this allows us to find integral representations for a functionals using standard techniques taken from pollaczek 1951

on the relation between orthogonal symplectic and unitary matrix ensembles for the unitary ensembles of n n hermitian matrices associated with a weight function w there is a kernel expressible in terms of the polynomials orthogonal with respect to the weight function which plays an important role for the orthogonal and symplectic ensembles of hermitian matrices there are 2 2 matrix kernels usually constructed using skew orthogonal polynomials which play an analogous role these matrix kernels are determined by their upper left hand entries we derive formulas expressing these entries in terms of the scalar kernel for the corresponding unitary ensembles we also show that whenever w w is a rational function the entries are equal to the scalar kernel plus some extra terms whose number equals the order of w w general formulas are obtained for these extra terms we do not use skew orthogonal polynomials in the derivations

infinite systems of noncolliding generalized meanders and riemann liouville differintegrals yor s generalized meander is a temporally inhomogeneous modification of the 2 1 dimensional bessel process with 1 in which the inhomogeneity is indexed by kappa in 0 2 nu 1 we introduce the noncolliding particle systems of the generalized meanders and prove that they are pfaffian processes in the sense that any multitime correlation function is given by a pfaffian in the infinite particle limit we show that the elements of matrix kernels of the obtained infinite pfaffian processes are generally expressed by the riemann liouville differintegrals of functions comprising the bessel functions j used in the fractional calculus where orders of differintegration are determined by as special cases of the two parameters the present infinite systems include the quaternion determinantal processes studied by forrester nagao and honner and by nagao which exhibit the temporal transitions between the universality classes of random matrix theory

verblunsky coefficients and nehari sequences we are concerned with a rather unfamiliar condition in the theory of the orthogonal polynomials on the unit circle in general the szego function is determined by its modulus while the condition in question is that it is also determined by its argument or in terms of the function theory that the square of the szego function is rigid in prediction theory this is known as a spectral characterization of complete nondeterminacy for stationary processes studied by bloomfield jewel and hayashi 1983 going back to a small but important result in the work of levinson and mckean 1964 it is also related with the cerebrated result of adamyan arov and krein 1968 for the nehari problem and there is a one one correspondence between the verblunsky coefficients and the negatively indexed fourier coefficients of the phase factor of the szego function which we call a nehari sequence we presents some fundamental results on the correspondence including extensions of the strong szego and baxter s theorems

on the largest singular values of random matrices with independent cauchy entries we apply the method of determinants to study the distribution of the largest singular values of large m n real rectangular random matrices with independent cauchy entries we show that for a special one parametric class of statistics the properties of the largest singular values rescaled by a factor 1 m2n2 agree in the limit with the statistical properties of the poisson random point process with the intensity 1 x 3 2 and therefore are different from the tracy widom law among other corollaries of our method we show an interesting connection between the mathematical expectations of the determinants of the complex rectangular m n standard wishart ensemble and the real rectangular 2m 2n standard wishart ensemble

asymptotics for the partition function in two cut random matrix models we obtain large n asymptotics for the random matrix partition function z n v int mathbb r n prod i j x i x j 2 prod j 1 ne nv x j dx j in the case where v is a polynomial such that the random matrix eigenvalues accumulate on two disjoint intervals the two cut case we compute leading and sub leading terms in the asymptotic expansion for log zn v up to terms that are small as n to infty our approach is based on the explicit computation of the first terms in the asymptotic expansion for a quartic symmetric potential v afterwards we use deformation theory of the partition function and of the associated equilibrium measure to generalize our results to general two cut potentials v the asymptotic expansion of log zn v as n to infty contains terms that depend analytically on the potential v and that have already appeared in the literature in addition our method allows us to compute the v independent terms of the asymptotic expansion of log zn v which to the best of our knowledge had not appeared before in the literature we use rigorous orthogonal polynomial and riemann hilbert techniques which had to this point only been successful to compute asymptotics for the partition function in the one cut case

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

partition functions for matrix models and isomonodromic tau functions for one matrix models with polynomial potentials the explicit relationship between the partition function and the isomonodromic tau function for the 2 2 polynomial differential systems satisfied by the associated orthogonal polynomials is derived

generalized orthogonal polynomials discrete kp and riemann hilbert problems classically a single weight on an interval of the real line leads to moments orthogonal polynomials and tridiagonal matrices appropriately deforming this weight with times t t 1 t 2 leads to the standard toda lattice and tau functions expressed as hermitian matrix integrals this paper is concerned with a sequence of t perturbed weights rather than one single weight this sequence leads to moments polynomials and a fuller matrix evolving according to the discrete kp hierarchy the associated tau functions have integral as well as vertex operator representations among the examples considered we mention nested calogero moser systems concatenated solitons and m periodic sequences of weights the latter lead to 2m 1 band matrices and generalized orthogonal polynomials also arising in the context of a riemann hilbert problem we show the riemann hilbert factorization is tantamount to the factorization of the moment matrix into the product of a lower times upper triangular matrix

large n limit for the product of two coupled random matrices for a pair of coupled rectangular random matrices we consider the squared singular values of their product which form a determinantal point process we show that the limiting mean distribution of these squared singular values is described by the second component of the solution to a vector equilibrium problem this vector equilibrium problem is defined for three measures with an upper constraint on the first measure and an external field on the second measure we carry out the steepest descent analysis for a 4 times 4 matrix valued riemann hilbert problem which characterizes the correlation kernel and is related to mixed type multiple orthogonal polynomials associated with the modified bessel functions a careful study of the vector equilibrium problem combined with this asymptotic analysis ultimately leads to the aforementioned convergence result for the limiting mean distribution an explicit form of the associated spectral curve as well as local sine meijer g and airy universality results for the squared singular values considered

random matrix models with additional interactions it has been argued that despite remarkable success existing random matrix theories are not adequate at describing disordered conductors in the metallic regime due to the presence of certain two body interactions in the effective hamiltonian for the eigenvalues in addition to the standard logarithmic interaction that arises entirely from symmetry considerations we present a new method that allows exact solution of random matrix models with such additional two body interactions this should broaden the scope of random matrix models in general

the dependence on the monodromy data of the isomonodromic tau function the isomonodromic tau function defined by jimbo miwa ueno vanishes on the malgrange s divisor of generalized monodromy data for which a vector bundle is nontrivial or which is the same a certain riemann hilbert problem has no solution in their original work jimbo miwa ueno provided an algebraic construction of its derivatives with respect to isomonodromic times however the dependence on the generalized monodromy data i e monodromy representation and stokes parameters was not derived we fill the gap by providing a simpler and more general description in which all the parameters of the problem monodromy changing and monodromy preserving are dealt with at the same level we thus provide variational formulae for the isomonodromic tau function with respect to the generalized monodromy data the construction applies more generally given any sufficiently well behaved family of riemann hilbert problems rhp where the jump matrices depend arbitrarily on deformation parameters we can construct a one form not necessarily closed on the deformation space malgrange s differential defined off malgrange s divisor we then introduce the notion of discrete schlesinger transformation it means that we allow the solution of the rhp to have poles or zeros at prescribed point s even if is not closed its difference evaluated along the original solution and the transformed one is shown to be the logarithmic differential on the deformation space of a function as a function of the position of the points of the schlesinger transformation it yields a natural generalization of the sato formula for the baker akhiezer vector even in the absence of a tau function and it realizes the solution of the rhp as such ba vector some exemplifications in the setting of the painleve ii equation and finite toplitz hankel determinants are provided

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

harmonic analysis on the infinite dimensional unitary group and determinantal point processes the infinite dimensional unitary group u is the inductive limit of growing compact unitary groups u n in this paper we solve a problem of harmonic analysis on u stated in ol3 the problem consists in computing spectral decomposition for a remarkable 4 parameter family of characters of u these characters generate representations which should be viewed as analogs of nonexisting regular representation of u the spectral decomposition of a character of u is described by the spectral measure which lives on an infinite dimensional space of indecomposable characters the key idea which allows us to solve the problem is to embed into the space of point configurations on the real line without two points this turns the spectral measure into a stochastic point process on the real line the main result of the paper is a complete description of the processes corresponding to our concrete family of characters we prove that each of the processes is a determinantal point process that is its correlation functions have determinantal form with a certain kernel our kernels have a special integrable form and are expressed through the gauss hypergeometric function from the analytic point of view the problem of computing the correlation kernels can be reduced to a problem of evaluating uniform asymptotics of certain discrete orthogonal polynomials studied earlier by richard askey and peter lesky one difficulty lies in the fact that we need to compute the asymptotics in the oscillatory regime with the period of oscillations tending to 0 we do this by expressing the polynomials in terms of a solution of a discrete riemann hilbert problem and computing the nonoscillatory asymptotics of this solution from the point of view of statistical physics we study thermodynamic limit of a discrete log gas system an interesting feature of this log gas is that its density function is asymptotically equal to the characteristic function of an interval our point processes describe how different the random particle configuration is from the typical densely packed configuration in simpler situations of harmonic analysis on infinite symmetric groups and harmonic analysis of unitarily invariant measures on infinite hermitian matrices similar results were obtained in our papers bo1 bo2 bo4

fredholm determinants jimbo miwa ueno τ functions and representation theory the authors show that a wide class of fredholm determinants arising in the representation theory of big groups such as the infinite dimensional unitary group solve painleve equations their methods are based on the theory of integrable operators and the theory of riemann hilbert problems 2002 wiley periodicals inc

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

how to prevent pressure oscillations in multicomponent flow calculations a new numerical scheme for computing multicomponent flows is presented it is able to handle strong shocks is pressure oscillation free through the contact discontinuities and guarantees the posi tivity of mass fractions some numerical experiments on shock tubes indicate the convergence to the correct weak solution comparisons with classical finite volume schemes are also proposed

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

2 matrix versus complex matrix model integrals over the unitary group as triangular integrals we prove that the 2 hermitian matrix model and the complex matrix model obey the same loop equations and as a byproduct we find a formula for itzykzon zuber type integrals over the unitary group integrals over u n are rewritten as gaussian integrals over triangular matrices and then computed explicitly that formula is an efficient alternative to the former shatashvili s formula

distributions on partitions point processes and the hypergeometric kernel we study a 3 parametric family of stochastic point processes on the one dimensional lattice originated from a remarkable family of representations of the infinite symmetric group we prove that the correlation functions of the processes are given by determinantal formulas with a certain kernel the kernel can be expressed through the gauss hypergeometric function we call it the hypergeometric kernel in a scaling limit our processes approximate the processes describing the decomposition of representations mentioned above into irreducibles as we showed in previous works the correlation functions of these limit processes also have determinantal form with so called whittaker kernel we show that the scaling limit of the hypergeometric kernel is the whittaker kernel integrable operator as defined by its izergin korepin and slavnov we argue that the hypergeometric kernel can be considered as a kernel defining a discrete integrable operator we also show that the hypergeometric kernel degenerates for certain values of parameters to the christoffel darboux kernel for meixner orthogonal polynomials this fact is parallel to the degeneration of the whittaker kernel to the christoffel darboux kernel for laguerre polynomials

commuting difference operators spinor bundles and the asymptotics of orthogonal polynomials with respect to varying complex weights abstract the paper has three parts in the first part we apply the theory of commuting pairs of pseudo difference operators to the formal asymptotics of orthogonal polynomials using purely geometrical arguments we show heuristically that the asymptotics for large degrees of orthogonal polynomial with respect to varying weights is intimately related to certain spinor bundles on a hyperelliptic algebraic curve reproducing formulae appearing in the works of deift et al on the subject in the second part we show that given an arbitrary nodal hyperelliptic curve satisfying certain conditions of admissibility we can reconstruct a sequence of polynomials orthogonal with respect to semiclassical complex varying weights supported on several curves in the complex plane the strong asymptotics of these polynomials will be shown to be given by the spinors introduced in the first part using a riemann hilbert analysis in the third part we use strebel theory of quadratic differentials and the procedure of welding to reconstruct arbitrary admissible hyperelliptic curves as a result we can obtain orthogonal polynomials whose zeroes may become dense on a collection of jordan arcs forming an arbitrary forest of trivalent loop free trees

an algorithm for les of premixed compressible flows using the conditional moment closure model a novel numerical method has been developed to couple a recent high order accurate fully compressible upwind method with the conditional moment closure combustion model the governing equations turbulence modelling and numerical methods are presented in full the new numerical method is validated against direct numerical simulation dns data for a lean premixed methane slot burner although the modelling approaches are based on non premixed flames and hence not expected to be valid for a wide range of premixed flames the predicted flame is just 10 longer than that in the dns and excellent agreement of mean mass fractions conditional mass fractions and temperature is demonstrated this new numerical method provides a very useful framework for future application of cmc to premixed as well as non premixed combustion

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

ar and ma representation of partial autocorrelation functions with applications we prove a representation of the partial autocorrelation function pacf or the verblunsky coefficients of a stationary process in terms of the ar and ma coefficients we apply it to show the asymptotic behaviour of the pacf we also propose a new definition of short and long memory in terms of the pacf

order statistics and ginibre s ensembles the moduli of the eigenvalues at the edge of ginibre s complex and quaternion gaussian random matrix ensembles are shown to respond to a limit theorem identical in nature to that for independent identically distributed sequences this is a companion work to ref 15 in which the limit law for the scaled spectral radius of these ensembles was identified

on tau functions for orthogonal polynomials and matrix models let v be a real polynomial of even degree and let rho be the equilibrium probability measure for v with support s to that v x greeater than or equal to int 2log x y rho dy c for some constant c with equality on s then s is the union of finitely many boundd intervals with endpoints delta j and rho is given by an algebraic weight w x on s then the system of orthogonal polynomials for w gives rise to a system of differential equations known as the schlesinger equations this paper identifies the tau function of this system with the hankel determinant det int x j k rho dx the solutions of the magnus schlesinger equation are realised by a linear system which is used to compute the tau functions in terms of a gelfand levitan equation the tau function is associated with a potential q and a scattering problem for the schrodinger equation with potential q the paper describes cases where this is integrable in terms of the nonlinear fourier transform

noncolliding brownian motion and determinantal processes a system of one dimensional brownian motions bms conditioned never to collide with each other is realized as i dyson s bm model which is a process of eigenvalues of hermitian matrix valued diffusion process in the gaussian unitary ensemble gue and as ii the h transform of absorbing bm in a weyl chamber where the harmonic function h is the product of differences of variables the vandermonde determinant the karlin mcgregor formula gives determinantal expression to the transition probability density of absorbing bm we show from the karlin mcgregor formula if the initial state is in the eigenvalue distribution of gue the noncolliding bm is a determinantal process in the sense that any multitime correlation function is given by a determinant specified by a matrix kernel by taking appropriate scaling limits spatially homogeneous and inhomogeneous infinite determinantal processes are derived we note that the determinantal processes related with noncolliding particle systems have a feature in common such that the matrix kernels are expressed using spectral projections of appropriate effective hamiltonians on the common structure of matrix kernels continuity of processes in time is proved and general property of the determinantal processes is discussed

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

simple matrix representations of the orthogonal polynomials for a rational spectral density on the unit circle abstract in this note by using a discrete analog of a projection formula introduced by a seghier in 1978 we calculate the orthogonal polynomials on the unit circle for a rational spectral density having no zeros there and derive simple matrix representations of themselves their squared norms and the verblunsky coefficients

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

semiclassical orthogonal polynomials matrix models and isomonodromic tau functions the differential systems satisfied by orthogonal polynomials with arbitrary semiclassical measures supported on contours in the complex plane are derived as well as the compatible systems of deformation equations obtained from varying such measures these are shown to preserve the generalized monodromy of the associated rank 2 rational covariant derivative operators the corresponding matrix models consisting of unitarily diagonalizable matrices with spectra supported on these contours are analyzed and it is shown that all coefficients of the associated spectral curves are given by logarithmic derivatives of the partition function or more generally the gap probabilities the associated isomonodromic tau functions are shown to coincide within an explicitly computed factor with these partition functions

universal structure and universal equations pde for unitary ensembles random matrix ensembles with unitary invariance of measure ue are described in a unified way using a combination of tracy widom tw and adler shiota van moerbeke approaches to the derivation of partial differential equations pdes for spectral gap probabilities first general three term recurrence relations for ue restricted to subsets of real line or in other words for functions in the resolvent kernel are obtained using them simple universal relations between all tw dependent variables and one dimensional toda lattice functions are found a universal system of pde for ue is derived from previous relations which leads also to a single independent pde for spectral gap probability of various ue thus orthogonal function bases and toda lattice are seen at the core of correspondence of different approaches moreover toda akns system provides a common structure of pde for unitary ensembles interestingly this structure can be seen in two very different forms one arises from orthogonal functi