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

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

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

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

on the analysis of incomplete spectra in random matrix theory through an extension of the jimbo miwa ueno differential abstract several distribution functions in the classical unitarily invariant matrix ensembles are prime examples of isomonodromic tau functions as introduced by jimbo miwa and ueno jmu in the early 1980s 45 recent advances in the theory of tau functions 41 based on earlier works of b malgrange and m bertola have allowed to extend the original jimbo miwa ueno differential form to a 1 form closed on the full space of extended monodromy data of the underlying lax pairs this in turn has yielded a novel approach for the asymptotic evaluation of isomonodromic tau functions including the exact computation of all relevant constant factors we use this method to efficiently compute the tail asymptotics of soft edge hard edge and bulk scaled distribution and gap functions in the complex wishart ensemble provided each eigenvalue particle has been removed independently with probability 1 0 1

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

gap probability of the circular unitary ensemble with a fisher hartwig singularity and the coupled painleve v system we consider the circular unitary ensemble with a fisher hartwig singularity of both jump type and root type at z 1 a rescaling of the ensemble at the fisher hartwig singularity leads to the confluent hypergeometric kernel by studying the asymptotics of the toeplitz determinants we show that the probability of there being no eigenvalues in a symmetric arc about the singularity on the unit circle for a random matrix in the ensemble can be explicitly evaluated via an integral of the hamiltonian of the coupled painleve v system in dimension four this leads to a painleve type representation of the confluent hypergeometric kernel determinant moreover the large gap asymptotics including the constant terms are derived by evaluating the total integral of the hamiltonian in particular we reproduce the large gap asymptotics of the confluent hypergeometric kernel determinant obtained by deift krasovsky and vasilevska and the sine kernel determinant as a special case including the constant term conjectured earlier by dyson

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

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

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

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

dependence modelling for spatial extremes current dependence models for spatial extremes are based upon max stable processes within this class there are few inferentially viable models available and we propose one further model more problematic are the restrictive assumptions that must be made when using max stable processes to model dependence for spatial extremes it must be assumed that the dependence structure of the observed extremes is compatible with a limiting model that holds for all events more extreme than those that have already occurred this problem has long been acknowledged in the context of finite dimensional multivariate extremes in particular when data display dependence at observable levels but are independent in the limit we propose a flexible class of models that is suitable for such data in a spatial context in addition we consider the situation where the extremal dependence structure may vary with distance we apply our models to spatially referenced significant wave height data from the north sea finding evidence that their extremal structure is not compatible with a limiting dependence model

the discrete logarithm hides o log n bits the main result of this paper is that obtaining any information about the o log p most significant bits of x given g x bmod p even with a tiny advantage over guessing is equivalent to computing discrete logarithms bmod p

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

a new class of models for bivariate joint tails summary a fundamental issue in applied multivariate extreme value analysis is modelling dependence within joint tail regions the primary focus of this work is to extend the classical pseudopolar treatment of multivariate extremes to develop an asymptotically motivated representation of extremal dependence that also encompasses asymptotic independence starting with the usual mild bivariate regular variation assumptions that underpin the coefficient of tail dependence as a measure of extremal dependence our main result is a characterization of the limiting structure of the joint survivor function in terms of an essentially arbitrary non negative measure that must satisfy some mild constraints we then construct parametric models from this new class and study in detail one example that accommodates asymptotic dependence asymptotic independence and asymmetry within a straightforward parsimonious parameterization we provide a fast simulation algorithm for this example and detail likelihood based inference including tests for asymptotic dependence and symmetry which are useful for submodel selection we illustrate this model by application to both simulated and real data in contrast with the classical multivariate extreme value approach which concentrates on the limiting distribution of normalized componentwise maxima our framework focuses directly on the structure of the limiting joint survivor function and provides significant extensions of both the theoretical and the practical tools that are available for joint tail modelling

modelling across extremal dependence classes different dependence scenarios can arise in multivariate extremes entailing careful selection of an appropriate class of models in bivariate extremes the variables are either asymptotically dependent or are asymptotically independent most available statistical models suit one or other of these cases but not both resulting in a stage in the inference that is unaccounted for but can substantially impact subsequent extrapolation existing modelling solutions to this problem are either applicable only on sub domains or appeal to multiple limit theories we introduce a unified representation for bivariate extremes that encompasses a wide variety of dependence scenarios and applies when at least one variable is large our representation motivates a parametric model that encompasses both dependence classes we implement a simple version of this model and show that it performs well in a range of settings

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

universality for the focusing nonlinear schrodinger equation at the gradient catastrophe point rational breathers and poles of the tritronquee solution to painleve i the semiclassical zero dispersion limit of solutions to the one dimensional focusing nonlinear schrodinger equation nls is studied in a scaling neighborhood d of a point of gradient catastrophe x 0 t 0 we consider a class of solutions specified in the text that decay as the neighborhood d contains the region of modulated plane wave with rapid phase oscillations as well as the region of fast amplitude oscillations spikes in this paper we establish the following universal behaviors of the nls solutions q near the point of gradient catastrophe i each spike has height and uniform shape of the rational breather solution to the nls scaled to the size ii the location of the spikes is determined by the poles of the tritronquee solution of the painleve i p1 equation through an explicit map between d and a region of the painleve independent variable iii if but lies away from the spikes the asymptotics of the nls solution is given by the plane wave approximation with the correction term being expressed in terms of the tritronquee solution of p1 the relation with the conjecture of dubrovin grava and klein about the behavior of solutions to the focusing nls near a point of gradient catastrophe is discussed we conjecture that the p1 hierarchy occurs at higher degenerate catastrophe points and that the amplitudes of the spikes are odd multiples of the amplitude at the corresponding catastrophe point our technique is based on the nonlinear steepest descent method for matrix riemann hilbert problems and discrete schlesinger isomonodromic transformations 2013 wiley periodicals inc

gauss periods orders and cryptographical applications experimental results on the multiplicative orders of gauss periods in finite fields are presented these results indicate that gauss periods have high order and are often primitive self dual normal elements in finite fields it is shown that gauss periods can be exponentiated in quadratic time an application is an efficient pseudorandom bit generator

hankel determinant and orthogonal polynomials for a gaussian weight with a discontinuity at the edge we compute asymptotics for hankel determinants and orthogonal polynomials with respect to a discontinuous gaussian weight in a critical regime where the discontinuity is close to the edge of the associated equilibrium measure support their behavior is described in terms of the ablowitz segur family of solutions to the painleve ii equation our results complement the ones in 33 as consequences of our results we conjecture asymptotics for an airy kernel fredholm determinant and total integral identities for painleve ii transcendents and we also prove a new result on the poles of the ablowitz segur solutions to the painleve ii equation we also highlight applications of our results in random matrix theory

correlations of the characteristic polynomials in the gaussian unitary ensemble or a singular hankel determinant we obtain large n asymptotics for products of powers of the absolute values of the characteristic polynomials in the gaussian unitary ensemble gue of n n matrices our results can also be interpreted as asymptotics of the determinant of a hankel matrix whose symbol is supported on the real line and possesses power like fisher hartwig singularities

asymptotics of toeplitz hankel and toeplitz hankel determinants with fisher hartwig singularities we study the asymptotics in n for n dimensional toeplitz determinants whose symbols possess fisher hartwig singularities on a smooth background we prove the general nondegenerate asymptotic behavior as conjectured by basor and tracy we also obtain asymptotics of hankel determinants on a nite interval as well as determinants of toeplitz hankel type our analysis is based on a study of the related system of orthogonal polynomials on the unit circle using the riemann hilbert approach

riemann hilbert problem associated with angelesco systems angelesco systems of measures with jacobi type weights are considered for such systems strong asymptotics for the related multiple orthogonal polynomials are found as well as the szego type functions in the procedure an approach from the riemann hilbert problem plays a fundamental role

large gap asymptotics for airy kernel determinants with discontinuities we obtain large gap asymptotics for airy kernel fredholm determinants with any number m of discontinuities these m point determinants are generating functions for the airy point process and encode probabilistic information about eigenvalues near soft edges in random matrix ensembles our main result is that the m point determinants can be expressed asymptotically as the product of m 1 point determinants multiplied by an explicit constant pre factor which can be interpreted in terms of the covariance of the counting function of the process

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

asymptotics of the airy kernel determinant the authors use riemann hilbert methods to compute the constant that arises in the asymptotic behavior of the airy kernel determinant of random matrix theory

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

the nonlinear steepest descent approach to the singular asymptotics of the second painleve transcendent abstract we consider the real valued solutions of the second painleve equation on the real line the two parameter family of the solutions having singular asymptotics as x or and x is studied with the help of the deift zhou nonlinear steepest descent method explicit evaluation in terms of trigonometric functions of the singular leading orders of the asymptotics is carried out and the corresponding connection formulae obtained a novel methodological feature is the appearance of the soliton type riemann hilbert problem in the course of the implementation of the deift zhou scheme for the riemann hilbert problem corresponding to the second painleve equation within the riemann hilbert isomonodromy approach the result of the paper reproduces previously known formulae derived by kapaev via the original isomonodromy technique

models and inference for uncertainty in extremal dependence conventionally modelling of multivariate extremes has been based on the class of multivariate extreme value distributions more recently other classes have been developed allowing for the possibility that whilst dependence is observed at finite levels the limit distribution is independent a number of articles have shown this development to be important for accurate estimation of the extremal properties both of theoretical processes and observed datasets it has also been shown that so far as dependence is concerned the choice between modelling with either asymptotically dependent or asymptotically independent distributions can be far more influential than model choice within either of these two classes in this paper we explore the issue of modelling across both classes examining in particular the effect of uncertainty caused by lack of knowledge about the status of asymptotic dependence this is achieved by new multivariate models whose parameter spaces are such that asymptotic dependence occurs on a boundary standard techniques in bayesian inference implemented through markov chain monte carlo enable inferences to be drawn that assign posterior probability mass to the boundary region the techniques are illustrated on a set of oceanographic data for which previous analyses have shown that it is difficult to resolve the question of asymptotic dependence status which is however important in model extrapolation

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

asymptotics of tracy widom distributions and the total integral of a painleve ii function the tracy widom distribution functions involve integrals of a painleve ii function starting from positive infinity in this paper we express the tracy widom distribution functions in terms of integrals starting from minus infinity there are two consequences of these new representations the first is the evaluation of the total integral of the hastings mcleod solution of the painleve ii equation the second is the evaluation of the constant term of the asymptotic expansions of the tracy widom distribution functions as the distribution parameter approaches minus infinity for the gue tracy widom distribution function this gives an alternative proof of the recent work of deift its and krasovsky the constant terms for the goe and gse tracy widom distribution functions are new

deformations of the tracy widom distribution in random matrix theory the tracy widom tw distribution describes the behavior of the largest eigenvalue we consider here two models in which tw undergoes transformations in the first one disorder is introduced in the gaussian ensembles by superimposing an external source of randomness a competition between tw and a normal gaussian distribution results depending on the spreading of the disorder the second model consists of removing at random a fraction of correlated eigenvalues of a random matrix the usual formalism of fredholm determinants extends naturally a continuous transition from tw to the weilbull distribution characteristic of extreme values of an uncorrelated sequence is obtained

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

modeling spatial processes with unknown extremal dependence class abstractmany environmental processes exhibit weakening spatial dependence as events become more extreme well known limiting models such as max stable or generalized pareto processes cannot capture this which can lead to a preference for models that exhibit a property known as asymptotic independence however weakening dependence does not automatically imply asymptotic independence and whether the process is truly asymptotically in dependent is usually far from clear the distinction is key as it can have a large impact upon extrapolation that is the estimated probabilities of events more extreme than those observed in this work we present a single spatial model that is able to capture both dependence classes in a parsimonious manner and with a smooth transition between the two cases the model covers a wide range of possibilities from asymptotic independence through to complete dependence and permits weakening dependence of extremes even under asymptotic dependence censored likelihood based

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

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

an isomonodromy cluster of two regular singularities we consider a linear 2 2 matrix ode with two coalescing regular singularities this coalescence is restricted with an isomonodromy condition with respect to the distance between the merging singularities in a way consistent with the ode in particular a zero distance limit for the ode exists the monodromy group of the limiting ode is calculated in terms of the original one this coalescing process generates a limit for the corresponding nonlinear systems of isomonodromy deformations in our main example the latter limit reads as p 6 p 5 where p n is the nth painleve equation we also discuss some general problems which arise while studying the above mentioned limits for the painleve equations

double scaling limit in the random matrix model the riemann hilbert approach we prove the existence of the double scaling limit in the unitary matrix model with quartic interaction and we show that the correlation functions in the double scaling limit are expressed in terms of the integrable kernel determined by the function for the hastings mcleod solution to the painleve ii equation the proof is based on the riemann hilbert approach and the central point of the proof is an analysis of analytic semiclassical asymptotics for the function at the critical point in the presence of four coalescing turning points 2003 wiley periodicals inc

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

universality for eigenvalue correlations from the unitary ensemble associated with a family of singular weights we study the asymptotic behavior of the eigenvalue correlations for the unitary ensemble associated with a family of singular weights w x exp 1 x2 x 1 1 for 0 when 0 1 2 these are szego class weights and are non szego when 1 2 it is proved that the behavior in the bulk of the spectrum is described in terms of the sine kernel which persists the so called universality results while the local behavior at the edge of the spectrum is described in terms of the airy kernel a specific scaling of the limit reflects the singular behavior of orthogonal polynomials on 1 1 with respect to the weight w x

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

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 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

universality for orthogonal and symplectic laguerre type ensembles we give a proof of the universality conjecture for orthogonal 1 and symplectic 4 random matrix ensembles of laguerre type in the bulk of the spectrum as well as at the hard and soft spectral edges our results are stated precisely in the introduction theorems 1 1 1 4 1 6 and corollaries 1 2 1 5 1 7 they concern the appropriately rescaled kernels k n correlation and cluster functions gap probabilities and the distributions of the largest and smallest eigenvalues corresponding results for unitary 2 laguerre type ensembles have been proved by the fourth author in ref 23 the varying weight case at the hard spectral edge was analyzed in ref 13 for 2 in this paper we do not consider varying weights our proof follows closely the work of the first two authors who showed in refs 7 8 analogous results for hermite type ensembles as in refs 7 8 we use the version of the orthogonal polynomial method presented in refs 22 25 to analyze the local eigenvalue statistics the necessary asymptotic information on the laguerre type orthogonal polynomials is taken from ref 23

a conditional approach for multivariate extreme values with discussion summary multivariate extreme value theory and methods concern the characterization estimation and extrapolation of the joint tail of the distribution of a d dimensional random variable existing approaches are based on limiting arguments in which all components of the variable become large at the same rate this limit approach is inappropriate when the extreme values of all the variables are unlikely to occur together or when interest is in regions of the support of the joint distribution where only a subset of components is extreme in practice this restricts existing methods to applications where d is typically 2 or 3 under an assumption about the asymptotic form of the joint distribution of a d dimensional random variable conditional on its having an extreme component we develop an entirely new semiparametric approach which overcomes these existing restrictions and can be applied to problems of any dimension we demonstrate the performance of our approach and its advantages over existing methods by using theoretical examples and simulation studies the approach is used to analyse air pollution data and reveals complex extremal dependence behaviour that is consistent with scientific understanding of the process we find that the dependence structure exhibits marked seasonality with ex tremal dependence between some pollutants being significantly greater than the dependence at non extreme levels

tracy widom distributions in critical unitary random matrix ensembles and the coupled painleve ii system we study fredholm determinants of the painleve ii and painleve xxxiv kernels in certain critical unitary random matrix ensembles these determinants describe special gap probabilities of eigenvalues we obtain tracy widom formulas for the fredholm determinants which are explicitly given in terms of integrals involving a family of distinguished solutions to the coupled painleve ii system in dimension four moreover the large gap asymptotics for these fredholm determinants are derived where the constant terms are given explicitly in terms of the riemann zeta function

the security of all rsa and discrete log bits we study the security of individual bits in an rsa encrypted message en x we show that given en x predicting any single bit in x with only a nonnegligible advantage over the trivial guessing strategy is through a polynomial time reduction as hard as breaking rsa moreover we prove that blocks of o log log n bits of x are computationally indistinguishable from random bits the results carry over to the rabin encryption scheme considering the discrete exponentiation function gx modulo p with probability 1 o 1 over random choices of the prime p the analog results are demonstrated the results do not rely on group representation and therefore applies to general cyclic groups as well finally we prove that the bits of ax b modulo p give hard core predicates for any one way function f all our results follow from a general result on the chosen multiplier hidden number problem given an integer n and access to an algorithm px that on input a random a i zn returns a guess of the ith bit of ax mod n recover x we show that for any i if px has at least a nonnegligible advantage in predicting the ith bit we either recover x or obtain a nontrivial factor of n in polynomial time the result also extends to prove the results about simultaneous security of blocks of o log log n bits

on integrals of the tronquee solutions and the associated hamiltonians for the painleve ii equation abstract we consider a family of tronquee solutions of the painleve ii equation q s 2 q s 3 s q s 2 1 2 1 2 which is characterized by the stokes multipliers s 1 e 2 i s 2 s 3 e 2 i with being a free parameter these solutions include the well known generalized hastings mcleod solution as a special case if 0 we derive asymptotics of integrals of the tronquee solutions and the associated hamiltonians over the real axis for 1 2 and 0 with the constant terms evaluated explicitly our results agree with those already known in the literature if the parameters and are chosen to be special values some applications of our results in random matrix theory are also discussed

2 parameter tau τ function for the first painleve equation topological recursion and direct monodromy problem via exact wkb analysis we show that a 2 parameter family of tau functions for the first painleve equation can be constructed by the discrete fourier transform of the topological recursion partition function for a family of elliptic curves we also perform an exact wkb theoretic computation of the stokes multipliers of associated isomonodromy system assuming certain conjectures

universality of a double scaling limit near singular edge points in random matrix models we consider unitary random matrix ensembles z n s t 1 e n tr v s t m dm on the space of hermitian n n matrices m where the confining potential vs t is such that the limiting mean density of eigenvalues as n and s t 0 vanishes like a power 5 2 at a singular endpoint of its support the main purpose of this paper is to prove universality of the eigenvalue correlation kernel in a double scaling limit the limiting kernel is built out of functions associated with a special solution of the pi2 equation which is a fourth order analogue of the painleve i equation in order to prove our result we use the well known connection between the eigenvalue correlation kernel and the riemann hilbert rh problem for orthogonal polynomials together with the deift zhou steepest descent method to analyze the rh problem asymptotically the key step in the asymptotic analysis will be the construction of a parametrix near the singular endpoint for which we use the model rh problem for the special solution of the pi2 equation

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

uniform asymptotics of the pollaczek polynomials via the riemann hilbert approach the pollaczek weight is an example of the non szeg class in this paper we investigate the asymptotics of the pollaczek polynomials via the riemannhilbert approach in the analysis the original e

a jacobi type christoffel darboux formula for multiple orthogonal polynomials of mixed type abstract an alternative expression for the christoffel darboux formula for multiple orthogonal polynomials of mixed type is derived from the lu factorization of the moment matrix of a given measure and two sets of weights we use the action of the generalized jacobi matrix j also responsible for the recurrence relations on the linear forms and their duals to obtain the result

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

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

generic critical points of normal matrix ensembles the evolution of the degenerate complex curve associated with the ensemble at a generic critical point is related to the finite time singularities of laplacian growth it is shown that the scaling behaviour at a critical point of singular geometry x3 y2 is described by the first painleve transcendent the regularization of the curve resulting from discretization is discussed

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

large deformations of the tracy widom distribution i non oscillatory asymptotics we analyze the left tail asymptotics of deformed tracy widom distribution functions describing the fluctuations of the largest eigenvalue in invariant random matrix ensembles after removing each soft edge eigenvalue independently with probability 1 gamma in 0 1 as gamma varies a transition from tracy widom statistics gamma 1 to classical weibull statistics gamma 0 was observed in the physics literature by bohigas et al phys rev e 79 031117 2009 we provide a description of this transition by rigorously computing the leading order left tail asymptotics of the thinned goe gue and gse tracy widom distributions in this paper we obtain the asymptotic behavior in the non oscillatory region with gamma in 0 1 fixed for the goe gue and gse distributions and gamma uparrow 1 at a controlled rate for the gue distribution this is the first step in an ongoing program to completely describe the transition between tracy widom and weibull statistics as a corollary to our results we obtain a new total integral formula involving the ablowitz segur solution to the second painleve equation