moment matrices and multi component kp with applications to random matrix theory questions on random matrices and non intersecting brownian motions have led to the study of moment matrices with regard to several weights the main result of this paper is to show that the determinants of such moment matrices satisfy upon adding one set of time deformations for each weight the multi component kp hierarchy these determinants are thus tau functions for these integrable hierarchies the tau functions so obtained with appropriate shifts of the time parameters forward and backwards will be expressed in terms of multiple orthogonal polynomials for these weights and their cauchy transforms the main result is a vast generalization of a known fact about infinitesimal deformations of orthogonal polynomials it concerns an identity between the orthogonality of polynomials on the real line the bilinear identity in kp theory and a generating functional for the full kp theory an additional fact not discussed in this paper is that these functions satisfy virasoro constraints with respect to these time parameters

a numerical study of the small dispersion limit of the korteweg de vries equation and asymptotic solutions abstract we study numerically the small dispersion limit for the korteweg de vries kdv equation u t 6 u u x 2 u x x x 0 for 1 and give a quantitative comparison of the numerical solution with various asymptotic formulae for small in the whole x t plane the matching of the asymptotic solutions is studied numerically

a critical phenomenon in the two matrix model in the quartic quadratic case we study a critical behavior for the eigenvalue statistics in the two matrix model in the quartic quadratic case for certain parameters the eigenvalue distribution for one of the matrices has a limit that vanishes like a square root in the interior of the support the main result of the paper is a new kernel that describes the local eigenvalue correlations near that critical point the kernel is expressed in terms of a 4 4 riemann hilbert problem related to the hastings mcleod solution of the painleve ii equation we then compare the new kernel with two other critical phenomena that appeared in the literature before first we show that the critical kernel that appears in case of quadratic vanishing of the limiting eigenvalue distribution can be retrieved from the new kernel by means of a double scaling limit second we briefly discuss the relation with the tacnode singularity in noncolliding brownian motions that was recently analyzed although the limiting density in that model also vanishes like a square root at a certain interior point the process at the local scale is different from the process that we obtain in the two matrix model

random hermitian matrices and gaussian multiplicative chaos n berestycki s work is supported by epsrc grants ep l018896 1 and ep i03372x 1 m d wong is a phd student at the cambridge centre for analysis supported by epsrc grant ep l016516 1 some of this work was carried out while the first and third authors visited the university of helsinki funded in part by epsrc grant ep l018896 1 they also wish to thank the university of helsinki for its hospitality during this visit c webb wishes to thank the isaac newton institute for mathematical sciences for its hospitality during the random geometry program during which this project was initiated c webb was supported by the eemil aaltonen foundation grant stochastic dynamics on large random graphs and academy of finland grants 288318 and 308123

painleve equations nonlinear special functions the six painleve equations pi pvi were first discovered about a hundred years ago by painleve and his colleagues in an investigation of nonlinear second order ordinary differential equations recently there has been considerable interest in the painleve equations primarily due to the fact that they arise as reductions of the soliton equations which are solvable by inverse scattering consequently the painleve equations can be regarded as completely integrable equations and possess solutions which can be expressed in terms of solutions of linear integral equations despite being nonlinear equations although first discovered from strictly mathematical considerations the painleve equations have arisen in a variety of important physical applications including statistical mechanics plasma physics nonlinear waves quantum gravity quantum field theory general relativity nonlinear optics and fibre optics the painleve equations may be thought of a nonlinear analogues of the classical special functions they possess hierarchies of rational solutions and one parameter families of solutions expressible in terms of the classical special functions for special values of the parameters further the painleve equations admit symmetries under affine weyl groups which are related to the associated backlund transformations in this paper i discuss some of the remarkable properties which the painleve equations possess including connection formulae backlund transformations associated discrete equations and hierarchies of exact solutions in particular the second painleve equation pii is used to illustrate these properties and some of the applications of pii are also discussed

orthogonal laurent polynomials a new algebraic approach abstract in this paper some important algebraic aspects in the theory of orthogonal laurent polynomials such as the three term recurrence relation the christoffel darboux identity or the liouville ostrogradski formula are revisited from the riemann hilbert window these topics are considered for general ordered laurent polynomial sequences and not only for the usual balanced cases in addition a connection with szego polynomials orthogonal polynomials in the unit circle is explored

a christoffel darboux formula and a favard s theorem for orthogonal laurent polynomials on the unit circle let f k z k 0 be the family of orthonormal laurent polynomials on the unit circle which spans d in the ordering induced by p n e n 1 2 from the three term recurrence relation satisfied by f k z k 0 we deduce a christoffel darboux formula particular examples are considered and a favard type theorem is proved a connection with the ordering induced by p n e n 2 is also established

another quadrature rule of highest algebraic degree of precision we consider certain quadrature rules of highest algebraic degree of precision that involve strong stieltjes distributions i e strong distributions on the positive real axis the behavior of the parameters of these quadrature rules when the distributions are strong c inversive stieltjes distributions is given a quadrature rule whose parameters have explicit expressions for their determination is presented an application of this quadrature rule for the evaluation of a certain type of integrals is also given

clustering methods based on variational analysis in the space of measures we formulate clustering as a minimisation problem in the space of measures by modelling the cluster centres as a poisson process with unknown intensity function we derive a ward type clustering criterion which under the poisson assumption can easily be evaluated explicitly in terms of the intensity function we show that asymptotically i e for increasing total intensity the optimal intensity function is proportional to a dimension dependent power of the density of the observations for fixed finite total intensity no explicit solution seems available however the ward type criterion to be minimised is convex in the intensity function so that the steepest descent method of molchanov and zuyev 2001 can be used to approximate the global minimum it turns out that the gradient is similar in form to the functional to be optimised if we discretise over a grid the steepest descent algorithm at each iteration step increases the current intensity function at those points where the gradient is minimal at the expense of regions with a large gradient value the algorithm is applied to a toy one dimensional example a simulation from a popular spatial cluster model and a real life dataset from strauss 1975 concerning the positions of redwood seedlings finally we discuss the relative merits of our approach compared to classical hierarchical and partition clustering techniques as well as to modern model based clustering methods using markov point processes and mixture distributions

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

orthogonal laurent polynomials and strong moment theory a survey these topics are found in many parts of 20th century mathematics and its applications in mathematical physics chemistry statistics and engineering historically the analytic theory of continued fractions has played a central role in both the origin and the development of the other topics continued fractions are intimately related to pad e approximants and special functions emphasis is given to the development of strong moment theory and orthogonal laurent polynomials and to the related continued fractions quadrature formulas integral transforms and linear functionals by a strong moment problem we mean the following for a given bisequence n n of real numbers does there exist a distribution function such that

strong asymptotics for cauchy biorthogonal polynomials with application to the cauchy two matrix model we apply the nonlinear steepest descent method to a class of 3 3 riemann hilbert problems introduced in connection with the cauchy two matrix random model the general case of two equilibrium measures supported on an arbitrary number of intervals is considered in this case we solve the riemann hilbert problem for the outer parametrix in terms of sections of a spinorial line bundle on a three sheeted riemann surface of arbitrary genus and establish strong asymptotic results for the cauchy biorthogonal polynomials

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

differential systems for biorthogonal polynomials appearing in 2 matrix models and the associated riemann hilbert problem we consider biorthogonal polynomials that arise in the study of a generalization of two matrix hermitian models with two polynomial potentials v1 x v2 y of any degree with arbitrary complex coefficients finite consecutive subsequences of biorthogonal polynomials windows of lengths equal to the degrees of the potentials v1 and v2 satisfy systems of ode s with polynomial coefficients as well as pde s deformation equations with respect to the coefficients of the potentials and recursion relations connecting consecutive windows a compatible sequence of fundamental systems of solutions is constructed for these equations the stokes sectorial asymptotics of these fundamental systems are derived through saddle point integration and the riemann hilbert problem characterizing the differential equations is deduced

explicit invariant measures for products of random matrices we construct explicit invariant measures for a family of infinite products of random independent identically distributed elements of sl 2 c the matrices in the product are such that one entry is gamma distributed along a ray in the complex plane when the ray is the positive real axis the products are those associated with a continued fraction studied by letac seshadri z wahr verw geb 62 1983 485 489 who showed that the distribution of the continued fraction is a generalised inverse gaussian we extend this result by finding the distribution for an arbitrary ray in the complex right half plane and thus compute the corresponding lyapunov exponent explicitly when the ray lies on the imaginary axis the matrices in the infinite product coincide with the transfer matrices associated with a one dimensional discrete schrodinger operator with a random gamma distributed potential hence the explicit knowledge of the lyapunov exponent may be used to estimate the exponential rate of localisation of the eigenstates

optimal approximation and quantisation abstract a number of interesting functionals f x of a finite point set x in the euclidean space can be represented as integrals of a function that depends on the integration variable y and x restricted onto a certain set c y x that is determined by y and x and satisfies separation and uniform boundedness conditions for instance c y x can be the voronoi cell generated by x that contains y we single out the general properties of c and that ensure that the normalised infimum of f x over all sets x with cardinality n converges to a limit that can be identified using a particular form of the considered functionals include those that appear in quantisation problems for probability measures and in finding the optimal approximation of a function by splines tangent planes and triangulated surfaces for instance it is shown that the minimum l approximation error normalised by n of a sufficiently smooth bivariate convex function f using the convex triangulation converges to the l 1 1 norm of the determinant of the second derivative of f times a certain absolute constant

orthogonal laurent polynomials and continued fractions associated with log normal distributions abstract this paper describes properties and computational procedures related to orthogonal laurent polynomials continued fractions two point pade approximants strong moment problems and l gaussian quadrature associated with log normal distribution functions t defined by t q1 2 2 e lnt 2 2 0

variational analysis of functionals of poisson processes letf ii be a functional of a generally nonhomogeneous poisson process ii with intensity measure considering the expectatione f ii as a functional of from the cone m of positive finite measures we derive closed form expressions for its fra chet derivatives of all orders that generalize the perturbation analysis formulae for poisson processes variational methods developed for the space m allow us to obtain first and second order sufficient conditions for various types of constrained optimization problems fore f we study in detail optimization over the class of measures with a fixed total massa and develop a technique that often allows us to obtain the asymptotic behavior of the optimal intensity measure in the high intensity setting whena grows to infinity as a particular application we consider the problem of optimal placement of stations in the poisson model of a two layer telecommunication network

pade approximants of random stieltjes series for some measure s on the positive half line we study the convergence of the finite truncations of the continued fraction or equivalently of the diagonal padeapproximants of the function s using the dyson schmidt method for an equivalent one dimensional disordered system and the results of marklof et al we obtain explicit formulae in terms of modified bessel functions for the almost sure rate of convergence of these approximants and for the almost sure distribution of their poles

semiclassical expansions in the toda hierarchy and the hermitian matrix model an iterative algorithm for determining a class of solutions of the dispersionful 2 toda hierarchy characterized by string equations is developed this class includes the solution which underlies the large n limit of the hermitian matrix model in the one cut case it is also shown how the double scaling limit can be naturally formulated in this scheme

strong stieltjes distributions and orthogonal laurent polynomials with applications to quadratures and pade approximation starting from a strong stieltjes distribution general sequences of orthogonal laurent polynomials are introduced and sonic of their most relevant algebraic properties are studied from this perspective the connection between certain quadrature formulas associated with the distribution and two point pade approximants to the stieltjes transform of is revisited finally illustrative numerical examples are discussed

nonlinear equations for the recurrence coefficients of discrete orthogonal polynomials we study polynomials orthogonal on a uniform grid we show that each weight function gives two potentials and each potential leads to a structure relation lowering operator these results are applied to derive second order difference equations satisfied by the orthogonal polynomials and nonlinear difference equations satisfied by the recursion coefficients in the three term recurrence relations

two point pade expansions for a family of analytic functions abstract each member g z of a family of analytic functions defined by stieltjes transforms is shown to be represented by a positive t fraction the approximants of which form the main diagonal in the two point pade table of g z the positive t fraction is shown to converge to g z throughout a domain d a b z z b a uniformly on compact subsets in addition truncation error bounds are given for the approximants of the continued function these bounds supplement previously known bounds and apply in part of the domain of g z not covered by other bounds the proofs of our results employ properties of orthogonal l polynomials laurent polynomials and l gaussian quadrature which are of some interest in themselves a number of examples are considered

orthogonal laurent polynomials and two point pade approximants associated with dawson s integral starting from the coefficients of the power expansions for the dawson s integral both around the origin and infinity a linear functional acting on the space of laurent polynomials is defined in this paper properties of this functional are studied in connection with certain sequences of orthogonal laurent polynomials and two point pade approximants

multi critical unitary random matrix ensembles and the general painleve ii equation we study unitary random matrix ensembles of the form z n detm 2 e ntrvwdm where a 1 2 and v is such that the limiting mean eigenvalue density for n n oc and n n 1 vanishes quadratically at the origin in order to compute the double scaling limits of the eigenvalue correlation kernel near the origin we use the deift zhou steepest descent method applied to the riemann hilbert problem for orthogonal polynomials on the real line with respect to the weight x 2ae nv x here the main focus is on the construction of a local parametrix near the origin with functions associated with a special solution qa of the painleve ii equation q sq 2q3 a we show that qa has no real poles for a 1 2 by proving the solvability of the corresponding riemann hilbert problem we also show that the asymptotics of the recurrence coefficients of the orthogonal polynomials can be expressed in terms of qa in the double scaling limit

a riemann hilbert problem for biorthogonal polynomials we characterize the biorthogonal polynomials that appear in the theory of coupled random matrices via a riemann hilbert problem our riemann hilbert problem is different from the ones that were proposed recently by ercolani and mclaughlin kapaev and bertola et al we believe that our formulation may be tractable to asymptotic analysis

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

orthogonality and recurrence for ordered laurent polynomial sequences we consider orderings of nested subspaces of the space of laurent polynomials on the real line more general than the balanced orderings associated with the ordered bases 1 z 1 z z 2 z 2 and 1 z z 1 z 2 z 2 we show that with such orderings the sequence of orthonormal laurent polynomials determined by a positive linear functional satisfies a three term recurrence relation reciprocally we show that with such orderings a sequence of laurent polynomials which satisfies a recurrence relation of this form is orthonormal with respect to a certain positive functional

on the origin of integrability in matrix models the matrix integrals involved in 2d lattice gravity are studied at finiten the integrable systems that arise in the continuum theory are shown to result directly from the formulation of the matrix integrals in terms of orthogonal polynomials the partition function proves to be a tau function of the toda lattice hierarchy the associated linear problem is equivalent to finding the polynomial basis which diagonalizes the partition function the cases of one hermitian matrix one unitary matrix and hermitian matrix chains all fall within the toda framework

hermite pade approximants for a pair of cauchy transforms with overlapping symmetric supports hermite pade approximants of type ii are vectors of rational functions with a common denominator that interpolate a given vector of power series at infinity with maximal order we are interested in the situation when the approximated vector is given by a pair of cauchy transforms of smooth complex measures supported on the real line the convergence properties of the approximants are rather well understood when the supports consist of two disjoint intervals angelesco systems or two intervals that coincide under the condition that the ratio of the measures is a restriction of the cauchy transform of a third measure nikishin systems in this work we consider the case where the supports form two overlapping intervals in a symmetric way and the ratio of the measures extends to a holomorphic function in a region that depends on the size of the overlap we derive szeg type formulae for the asymptotics of the approximants identify the convergence and divergence domains the divergence domains appear for angelesco systems but are not present for nikishin systems and show the presence of overinterpolation a feature peculiar for nikishin systems but not for angelesco systems our analysis is based on a riemann hilbert problem for multiple orthogonal polynomials the common denominator 2016 wiley periodicals inc

multiple orthogonal polynomials of mixed type gauss borel factorization and the multi component 2d toda hierarchy multiple orthogonality is considered in the realm of a gauss borel factorization problem for a semi infinite moment matrix perfect combinations of weights and a finite borel measure are constructed in terms of m nikishin systems these perfect combinations ensure that the problem of mixed multiple orthogonality has a unique solution that can be obtained from the solution of a gauss borel factorization problem for a semi infinite matrix which plays the role of a moment matrix this leads to sequences of multiple orthogonal polynomials their duals and second kind functions it also gives the corresponding linear forms that are bi orthogonal to the dual linear forms expressions for these objects in terms of determinants from the moment matrix are given recursion relations are found which imply a multi diagonal jacobi type matrix with snake shape and results like the abc theorem or the christoffel darboux formula are re derived in this context using the factorization problem and the generalized hankel symmetry of the moment matrix the connection between this description of multiple orthogonality and the multi component 2d toda hierarchy which can be also understood and studied through a gauss borel factorization problem is discussed deformations of the weights natural for m nikishin systems are considered and the correspondence with solutions to the integrable hierarchy represented as a collection of lax equations is explored corresponding lax and zakharov shabat matrices as well as wave functions and their adjoints are determined the construction of discrete flows is discussed in terms of miwa transformations which involve darboux transformations for the multiple orthogonality conditions the bilinear equations are derived and the tau function representation of the multiple orthogonality is given

fractional brownian motion with hurst index h 0 and the gaussian unitary ensemble the goal of this paper is to establish a relation between characteristic polynomials of n n gue random matrices h as n and gaussian processes with logarithmic correlations we introduce a regularized version of fractional brownian motion with zero hurst index which is a gaussian process with stationary increments and logarithmic increment structure then we prove that this process appears as a limit of d n z log det h zi on mesoscopic scales as n by employing a fourier integral representation we use this to prove a continuous analogue of a result by diaconis and shahshahani j appl probab 31a 1994 49 62 on the macroscopic scale d n x gives rise to yet another type of gaussian process with logarithmic correlations we give an explicit construction of the latter in terms of a chebyshev fourier random series

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

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

on the convergence of certain gauss type quadrature formulas for unbounded intervals we consider the convergence of gauss type quadrature formulas for the integral f 0 f x x dx where w is a weight function on the half line 0 the n point gauss type quadrature formulas are constructed such that they are exact in the set of laurent polynomials p q 1 k p q 1 p a k x k where p p n is a sequence of integers satisfying 0 p n 2n and q q n 2n p n it is proved that under certain carleman type conditions for the weight and when p n or q n goes to oo then convergence holds for all functions f for which fw is integrable on 0 some numerical experiments compare the convergence of these quadrature formulas with the convergence of the classical gauss quadrature formulas for the half line

critical edge behavior in the modified jacobi ensemble and the painleve v transcendents we study the jacobi unitary ensemble defined by the perturbed chebyshev weight w x 1 x2 12 tn2 x2 x 1 1 tn 1 the algebraic singularity coalesces with the end point of the support of the weight function namely tn 1 the hard edge as n we use the riemann hilbert approach the method of deift and zhou to obtain the asymptotics of the polynomials orthogonal with respect to w x we show that the universality property is preserved the asymptotic behavior of the eigenvalue correlations in the bulk of the spectrum is described in terms of the sine kernel the main result is on the local behavior at the edge of the spectrum the limit kernel at the edge when tn 1 o n 2 is described by the function for a specific solution of the painleve v equation we also show that when tn varies to a fixed t 1 the painleve v kernel transits to the bessel kernel j 12 on the other hand when tn approaches 1 the painleve v kernel can be approximated by another bessel kernel j 12

the error in linear interpolation at the vertices of a simplex a formula for the error in multivariate quasi interpolation which reproduces the linear polynomials is given from it sharp pointwise l infty bounds for the error in linear interpolation interpolation by linear polynomials to function values at the vertices of a simplex are obtained the corresponding envelope theorem giving the optimal recovery of functions is discussed

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

optimum quantization and its applications abstract minimum sums of moments or equivalently distortion of optimum quantizers play an important role in several branches of mathematics fejes toth s inequality for sums of moments in the plane and zador s asymptotic formula for minimum distortion in euclidean d space are the first precise pertinent results in dimension d 2 in this article these results are generalized in the form of asymptotic formulae for minimum sums of moments resp distortion of optimum quantizers on riemannian d manifolds and normed d spaces in addition we provide geometric and analytic information on the structure of optimum configurations our results are then used to obtain information on i the minimum distortion of high resolution vector quantization and optimum quantizers ii the error of best approximation of probability measures by discrete measures and support sets of best approximating discrete measures iii the minimum error of numerical integration formulae for classes of holder continuous functions and optimum sets of nodes iv best volume approximation of convex bodies by circumscribed convex polytopes and the form of best approximating polytopes and v the minimum isoperimetric quotient of convex polytopes in minkowski spaces and the form of the minimizing polytopes

on the solvability of painleve ii and iv we introduce a rigorous methodology for studying the riemann hilbert problems associated with certain integrable nonlinear ordinary differential equations for concreteness we investigate the painleve ii and painleve iv equations we show that the cauchy problems for these equations admit in general global meromorphic in t solutions furthermore for special relations among the monodromy data and for t on stokes lines these solutions are bounded for finite t

asymptotics for toeplitz determinants perturbation of symbols with a gap we study the determinants of toeplitz matrices as the size of the matrices tends to infinity in the particular case where the symbol has two jump discontinuities and tends to zero on an arc of the unit circle at a sufficiently fast rate we generalize an asymptotic expansion by widom indiana univ math j 21 277 283 1971 which was known for symbols supported on an arc we highlight applications of our results in the circular unitary ensemble and in the study of fredholm determinants associated to the sine kernel

painleve ii asymptotics near the leading edge of the oscillatory zone for the korteweg de vries equation in the small dispersion limit in the small dispersion limit solutions to the korteweg de vries equation develop an interval of fast oscillations after a certain time we obtain a universal asymptotic expansion for the korteweg de vries solution near the leading edge of the oscillatory zone up to second order corrections this expansion involves the hastings mcleod solution of the painleve ii equation we prove our results using the riemann hilbert approach c 2009 wiley periodicals inc

the fourth painleve equation and associated special polynomials in this article rational solutions and associated polynomials for the fourth painleve equation are studied these rational solutions of the fourth painleve equation are expressible as the logarithmic derivative of special polynomials the okamoto polynomials the structure of the roots of these okamoto polynomials is studied and it is shown that these have a highly regular structure the properties of the okamoto polynomials are compared and contrasted with those of classical orthogonal polynomials further representations are given of the associated rational solutions in the form of determinants through schur functions

random continued fractions with beta hypergeometric distribution in a recent paper statist probab lett 78 2008 1711 1721 it has been shown that certain random continued fractions have a density which is a product of a beta density and a hypergeometric function 2f1 in the present paper we fully exploit a formula due to thomae j reine angew math 87 1879 26 73 in order to generalize substantially the class of random continuous fractions with a density of the above form this involves the design of seven particular graphs infinite paths on them lead to random continued fractions with an explicit distribution a careful study about the set of five real parameters leading to a beta hypergeometric distribution is required relying on almost forgotten results mainly due to felix klein

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

a pde for non intersecting brownian motions and applications abstract consider n n 1 n 2 n p non intersecting brownian motions on the real line starting from the origin at t 0 with n i particles forced to reach p distinct target points i at time t 1 with 1 2 p this can be viewed as a diffusion process in a sector of r n this work shows that the transition probability that is the probability for the particles to pass through windows e k at times t k satisfies in a new set of variables a non linear pde which can be expressed as a near wronskian that is a determinant of a matrix of size p 1 with each row being a derivative of the previous except for the last column it is an interesting open question to understand those equations from a more probabilistic point of view as an application of these equations let the number of particles forced to the extreme points 1 and p tend to infinity keep the number of particles forced to intermediate points fixed inliers but let the target points themselves go to infinity according to a proper scale a new critical process appears at the point of bifurcation where the bulk of the particles forced to n depart from those going to n these statistical fluctuations near that point of bifurcation are specified by a kernel which is a rational perturbation of the pearcey kernel this work also shows that such equations are an essential tool in obtaining certain asymptotic results finally the paper contains a conjecture

the asymptotic behaviour of recurrence coefficients for orthogonal polynomials with varying exponential weights we consider orthogonal polynomials p n n x n 0 on the real line with respect to a weight w x e n v x and in particular the asymptotic behaviour of the coefficients a n n and b n n in the three term recurrence x p n n x p n 1 n x b n n p n n x a n n p n 1 n x for one cut regular v we show using the deift zhou method of steepest descent for riemann hilbert problems that the diagonal recurrence coefficients a n n and b n n have asymptotic expansions as n in powers of 1 n 2 and powers of 1 n respectively

an independence property for the product of gig and gamma laws matsumoto and yor have recently discovered an interesting transformation which preserves a bivariate probability measure which is a product of the generalized inverse gaussian gig and gamma distributions this paper is devoted to a detailed study of this phenomenon let x and y be two independent positive random variables we prove theorem 4 1 that u x y 1 and v x 1 x y 1 are independent if and only if there exists p a b 0 such that y is gamma distributed with shape parameter p and scale parameter 2a 1 and such that x has a gig distribution with parameters p a and b the direct part for a b was obtained in matsumoto and yor the result is partially extended theorem 5 1 to the case where x and y are valued in the cone v of symmetric positive definite r r real matrices as follows under a hypothesis of smoothness of densities we prove that u x y 1 and v x 1 x y 1 are independent if and only if there exists p r 1 2 and a and b in v such that y is wishart distributed with shape parameter p and scale parameter 2a 1 and such that x has a matrix gig distribution with parameters p a and b the direct result is also extended to singular wishart distributions theorem 3 1