asymptotic dimension and small cancellation for hierarchically hyperbolic spaces and groups we prove that all hierarchically hyperbolic groups have finite asymptotic dimension one application of this result is to obtain the sharpest known bound on the asymptotic dimension of the mapping class group of a finite type surface improving the bound from exponential to at most quadratic in the complexity of the surface we also apply the main result to various other hierarchically hyperbolic groups and spaces we also prove a small cancellation result namely if g is a hierarchically hyperbolic group h g is a suitable hyperbolically embedded subgroup and n h is sufficiently deep in h then g n is a relatively hierarchically hyperbolic group this new class provides many new examples to which our asymptotic dimension bounds apply along the way we prove new results about the structure of hhss for example the associated hyperbolic spaces are always obtained up to quasi isometry by coning off canonical coarse product regions in the original space generalizing a relation established by masur and minsky between the complex of curves of a surface and teichmuller space

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

asymptotics and integrable structures for biorthogonal polynomials associated to a random two matrix model abstract we give a rigorous construction of complete families of biorthonormal polynomials associated to a planar measure of the form e n v x w y 2 xy d x d y for polynomial v and w we are further able to show that the zeroes of these polynomials are all real and distinct a complex analytical construction of the biorthonormal polynomials is given in terms of a non local riemann hilbert problem which given our prior result provides an avenue for developing uniform asymptotics for the statistical distributions of these zeroes as n becomes large the biorthonormal polynomials considered here play a fundamental role in the analysis of certain random multi matrix models we show that the evolutions of the recursion matrices for the polynomials induced by linear deformations of v and w coincide with a semi infinite generalization of the completely integrable full kostant toda lattice this connection could be relevant for understanding aspects of scaling limits for the multi matrix model

generalized christoffel darboux formula for classical skew orthogonal polynomials we show that skew orthogonal functions defined with respect to jacobi weight wa b x 1 x a 1 x b a b 1 including the limiting cases of laguerre wa x xa e x a 1 and gaussian weight satisfy three term recursion relation in the quaternion space from this we derive generalized christoffel darboux gcd formulae for kernel functions arising in the study of the corresponding orthogonal and symplectic ensembles of random 2n 2n matrices using the gcd formulae we calculate the level densities and prove that in the bulk of the spectrum under appropriate scaling the eigenvalue correlations are universal we also provide evidence to show that there exists a mapping between skew orthogonal functions arising in the study of orthogonal and symplectic ensembles of random matrices

random matrices in 2d laplacian growth and operator theory since it was first applied to the study of nuclear interactions by wigner and dyson almost 60 years ago random matrix theory rmt has developed into a field of its own within applied mathematics and is now essential to many parts of theoretical physics from condensed matter to high energy the fundamental results obtained so far rely mostly on the theory of random matrices in one dimension the dimensionality of the spectrum or equilibrium probability density in the last few years this theory has been extended to the case where the spectrum is two dimensional or even fractal with dimensions between 1 and 2 in this paper we review these recent developments and indicate some physical problems where the theory can be applied

skew orthogonal polynomials differential systems and random matrix theory we study skew orthogonal polynomials with respect to the weight function exp 2v x with v x 2dk 1 uk k xk u2d 0 d 0 a finite subsequence of such skew orthogonal polynomials arising in the study of orthogonal and symplectic ensembles of random matrices satisfies a system of differential difference deformation equation the vectors formed by such subsequence have the rank equal to the degree of the potential in the quaternion sense these solutions satisfy certain compatibility condition and hence admit a simultaneous fundamental system of solutions

on smoothing and order reduction effects for implicit runge kutta formulae abstract it is well known that many important classes of runge kutta methods suffer an order reduction phenomenon when applied to certain classes of stiff problems in particular the s stage gauss methods with stage order s and order of consistency 2s behave like methods of order s when applied to the class of singularly perturbed problems in this paper we will show that the process of smoothing can ameliorate this effect when dealing with initial value problems by first studying the effect of smoothing on the standard prothero robinson problem and then by extending the analysis to the general class of singularly perturbed problems

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

random matrices virasoro algebras and noncommutative kp department of mathematics brandeis university waltham mass 02254 usa email adler math brandeis edu the support of a national science foundation grant dms 9503246 is gratefully acknowledged department of mathematics kyoto university kyoto 606 japan the hospitality of the university of louvain and brandeis university is gratefully acknowledged department of mathematics universite de louvain 1348 louvain la neuve belgium and brandeis university waltham mass 02254 usa e mail vanmoerbeke geom ucl ac be and math brandeis edu the support of a national science foundation grant dms 9503246 a nato a fnrs and a francqui foundation grant is gratefully acknowledged a corrected version has appeared in the duke mathematical journal 94 379 431 1998

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

deformation analysis of matrix models abstract in this paper we show that the non linear equations that play a role in tracy and widom s analysis of scaling limits for hermitian matrix models are special cases of the monodromy preserving deformation equations first discussed by sato miwa and jimbo in the early 1980 s

peripheral fillings of relatively hyperbolic groups in this paper a group theoretic version of dehn surgery is studied starting with an arbitrary relatively hyperbolic group g we define a peripheral filling procedure which produces quotients of g by imitating the effect of the dehn filling of a complete finite volume hyperbolic 3 manifold m on the fundamental group 1 m the main result of the paper is an algebraic counterpart of thurston s hyperbolic dehn surgery theorem we also show that peripheral subgroups of g almost have the congruence extension property and the group g is approximated in an algebraic sense by its quotients obtained by peripheral fillings

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

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

duality biorthogonal polynomials and multi matrix models the statistical distribution of eigenvalues of pairs of coupled random matrices can be expressed in terms of integral kernels having a generalized christoffel darboux form constructed from sequences of biorthogonal polynomials for measures involving exponentials of a pair of polynomials v1 v2 in two different variables these kernels may be expressed in terms of finite dimensional windows spanned by finite subsequences having length equal to the degree of one or the other of the polynomials v1 v2 the vectors formed by such subsequences satisfy dual pairs of first order systems of linear differential equations with polynomial coefficients having rank equal to one of the degrees of v1 or v2 and degree equal to the other they also satisfy recursion relations connecting the consecutive windows and deformation equations determining how they change under variations in the coefficients of the polynomials v1 and v2 viewed as overdetermined systems of linear difference differential deformation equations these are shown to be compatible and hence to admit simultaneous fundamental systems of solutions the main result is the demonstration of a spectral duality property namely that the spectral curves defined by the characteristic equations of the pair of matrices defining the dual differential systems are equal upon interchange of eigenvalue and polynomial parameters

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

partition function for multi cut matrix models we consider the partition function z n of a random matrix model with polynomial potential v t 1 t 2 2 t 2d 2d it is known that the second logarithmic derivative of z n with respect to the times t k can be expressed in terms of the recurrence coefficients of the related orthogonal polynomials an explicit formula for the recurrence coefficients of the orthogonal polynomials in the limit n oo for multi cut regular v has been derived in 10 through the riemann hilbert approach the expression for z n in the limit n has been derived in 7 through a mean field approach we show that the above asymptotic formulae satisfy the same relations that hold for finite n

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

tight geodesics in the curve complex the curve graph mathcal g associated to a compact surface is the 1 skeleton of the curve complex defined by harvey masur and minsky showed that this graph is hyperbolic and defined the notion of a tight geodesic therein we prove some finiteness results for such geodesics for example we show that a slice of the union of tight geodesics between any pair of points has cardinality bounded purely in terms of the topological type of we deduce some consequences for the action of the mapping class group on mathcal g in particular we show that it satisfies an acylindricity condition and that the stable lengths of pseudoanosov elements are rational with bounded denominator

asymptotic error expansions for stiff equations an analysis for the implicit midpoint and trapezoidal rules in the strongly stiff case the structure of the global discretization error is studied for the implicit midpoint and trapezoidal rules applied to nonlinearstiff initial value problems the point is that in general the global error contains nonsmooth oscillating terms at the dominanth 2 level however it is shown in the present paper that for special classes of stiff problems these nonsmooth terms contain an additional factor where 1 is the magnitude of the stiff eigenvalues in these cases a full asymptotic error expansion exists in thestrongly stiff case sufficiently small compared to the stepsizeh the general case where the oscillating error components areo h 2 and noto h 2 and applications of our results extrapolation and defect correction algorithims will be studied in separate papers

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

developments in random matrix theory in this introduction to the journal of physics a special issue on random matrix theory we give a review of the main historical developments in random matrix theory a short summary of the papers that appear in this special issue is also given

mixed correlation function and spectral curve for the 2 matrix model we compute the mixed correlation function in a way which involves only the orthogonal polynomials with degrees close to n in some sense like the christoffel darboux theorem for non mixed correlation functions we also derive new representations for the differential systems satisfied by the bi orthogonal polynomials and we find new formulae for the spectral curve in particular we prove the conjecture of m bertola claiming that the spectral curve is the same curve which appears in the loop equations

asymptotic error expansions for stiff equations the implicit euler scheme this paper discusses the existence of an asymptotic expansion for the global error of the implicit euler scheme applied to stiff nonlinear systems of ordinary differential equations it is shown that in strongly stiff situations a full asymptotic expansion exists at all gridpoints for the mildly stiff case it is shown that the full order of the remainder term which inevitably breaks down at the first gridpoints after a stepsize change reappears at the subsequent gridpoints our analysis is based on singular perturbation techniques

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

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

extrapolation at stiff differential equations asymptotic expansions of the global error of numerical methods are well understood if the differential equation is non stiff this paper is concerned with such expansions for the implicit euler method the linearly implicit euler method and the linearly implicit mid point rule when they are applied tostiff differential equations in this case perturbation terms are present whose dominant one is given explicitly this permits us to better understand the behaviour ofextrapolation methods at stiff differential equations numerical examples supporting the theoretical results are included

a method of integration over matrix variables the integral over twon n hermitan matricesz g c dadbexp tr a 2 b 2 2cab g n a 4 b 4 is evaluated in the limit of largen for this purpose use is made of the theory of diffusion equation and that of orthogonal polynomials with a non local weight the above integral arises in the study of the planar approximation to quantum field theory

semiclassical asymptotics of orthogonal polynomials riemann hilbert problem and universality in the matrix model we derive semiclassical asymptotics for the orthogonal polynomials pn z on the line with respect to the exponential weight exp inv z where v z is a double well quartic polynomial in the limit when n n 1 we assume that n n cri for some 0 where cr is the critical value which separates orthogonal polynomials with two cuts from the ones with one cut simultaneously we derive semiclassical asymptotics for the recursive coe cients of the orthogonal polynomials and we show that these coe cients form a cycle of period two which drifts slowly with the change of the ratio n n the proof of the semiclassical asymptotics is based on the methods of the theory of integrable systems and on the analysis of the appropriate matrix riemannhilbert problem as an application of the semiclassical asymptotics of the orthogonal polynomials we prove the universality of the local distribution of eigenvalues in the matrix model with the double well quartic interaction in the presence of two cuts contents

a hurewicz type theorem for asymptotic dimension and applications to geometric group theory we prove an asymptotic analog of the classical hurewicz theorem on mappings that lower dimension this theorem allows us to find sharp upper bound estimates for the asymptotic dimension of groups acting on finite dimensional metric spaces and allows us to prove a useful extension theorem for asymptotic dimension as applications we find upper bound estimates for the asymptotic dimension of nilpotent and polycyclic groups in terms of their hirsch length we are also able to improve the known upper bounds on the asymptotic dimension of fundamental groups of complexes of groups amalgamated free products and the hyperbolization of metric spaces possessing the higson property

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

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

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

the concept of b convergence a useful concept of convergence for nonlinear stiff ode s will be developed which permits the derivation of uniform global error bounds independent of the stiffness of the considered problem this concept will be discussed for three simple methods for stiff systems the implicit euler scheme the implicit midpoint rule and the implicit trapezoidal rule

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

the novikov conjecture for groups with finite asymptotic dimension recall that the asymptotic dimension is a coarse geometric analogue of the covering dimension in topology page 28 14 more precisely the asymptotic dimension for a metric space is the smallest integer n such that for any r 0 there exists a uniformly bounded cover c ui iei of the metric space for which the r multiplicity of c is at most n 1 i e no ball of radius r in the metric space intersects more than n 1 members of c 14 the class of finitely generated discrete groups with finite asymptotic dimension is hereditary in the sense that if a finitely generated group has finite asymptotic dimension as metric space with a word length metric then its finitely generated subgroups also have finite asymptotic dimension as metric spaces with word length metrics cf section 6 this together with a result of gromov in 14 implies that finitely generated subgroups of gromov s hyperbolic groups have finite asymptotic dimension currently no example of a finitely generated group with infinite asymptotic dimension and finite classifying space is known it should also be noted that two different definitions of asymptotic dimension

a semi implicit mid point rule for stiff systems of ordinary differential equations the paper introduces a new semi implicit extrapolation method especially designed for the numerical solution of stiff systems of ordinary differential equations the existence of a quadratic asymptotic expansion in terms of the stepsize is shown moreover the new discretization is analyzed in the light of well known stability models the efficiency of the new integrator is clearly demonstrated by solving a series of challenging test problems including real life examples