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

isospectral flows related to frobenius stickelberger thiele polynomials the isospectral deformations of the frobenius stickelberger thiele fst polynomials introduced in spiridonov et al commun math phys 272 139 165 2007 are studied for a specific choice of the deformation of the spectral measure one is led to an integrable lattice fst lattice which is indeed an isospectral flow connected with a generalized eigenvalue problem in the second part of the paper the spectral problem used previously in the study of the modified camassa holm mch peakon lattice is interpreted in terms of the fst polynomials together with the associated fst polynomials resulting in a map from the mch peakon lattice to a negative flow of the finite fst lattice furthermore it is pointed out that the degenerate case of the finite fst lattice unexpectedly maps to the interlacing peakon ode system associated with the two component mch equation studied in chang et al adv math 299 1 35 2016

the toda hierarchy and the kdv hierarchy this paper considers the relation between the periodic kdv hierarchy and the limit of the periodic toda hierarchies by choosing the initial data of the toda flows in a canonical way the behavior of a certain toda flow can mimic kdv flows conjecturally a method of deforming the kdv hierarchy is given

solutions of the bigraded toda hierarchy the n m bigraded toda hierarchy is an extension of the original toda lattice hierarchy the pair of numbers n m represents the band structure of the lax matrix which has n upper and m lower diagonals and the original one is referred to as the 1 1 bigraded toda hierarchy because of this band structure one can introduce m n 1 commuting flows which give a parametrization of a small phase space for a topological field theory in this paper first we show that there exists a natural symmetry between the n m and m n bigraded toda hierarchies we then derive the hirota bilinear form for those commuting flows which consist of two dimensional toda hierarchy the discrete kp hierarchy and its backlund transformations we also discuss the solution structure of the n m bigraded toda equation in terms of the moment matrix defined via the wave operators associated with the lax operator and construct some of the explicit solutions in particular we give the rational solutions which are expressed by the products of the schur polynomials corresponding to the non rectangular young diagrams

parabolic orbifolds and the dimension of the maximal measure for rational maps let f g g be a rational map of the riemann sphere d e g f 2 a natural invariant measure m the measure of maximal entropy was constructed by ljubich lju and independently by freire lopes and mafi6 flm the aim of this paper is to compare this measure with some hausdorff measures first recall the following definition for a probabili ty measure v on or more generally on a smooth manifold the hausdorff dimension of v is defined by a formula hd v inf hd y y v y l

on the christoffel darboux formula for generalized matrix orthogonal polynomials abstract we obtain an extension of the christoffel darboux formula for matrix orthogonal polynomials with a generalized hankel symmetry including the adler van moerbeke generalized orthogonal polynomials

the correspondence between tracy widom and adler shiota van moerbeke approaches in random matrix theory the gaussian case two approaches to the derivation of integrable differential equations for random matrix probabilities are compared both methods are rewritten in such a form that simple and explicit relations between all tw dependent variables and functions of asvm are found for the example of finite size gaussian matrices orthogonal function systems and toda lattice are seen as the core structure of both approaches and their relationship

the isomonodromy approach to matric models in 2d quantum gravity we consider the double scaling limit in the hermitian matrix model for 2d quantum gravity associated with the measure exp sum limits j 1 n t j z 2j n geqq 3 we show that after the appropriate modification of the contour of integration the cross migdal douglas shenker limit to the painleve i equation in the generic case of the pure gravity is valid and calculate the nonperturbative parameters of the corresponding painleve function our approach is based on the wkb analysis of the l a pair corresponding to the discrete string equation in the framework of the inverse monodromy method here we extend our results which were obtained before for the particular casesn 2 3 our analysis complements the isomonodromy approach proposed by g moore to the general string equations that come from the matrix model in the continuous limit and differ in that we apply the isomonodromy technique to investigate the double scaling limit itself

the spectrum of coupled random matrices the study of the spectrum of coupled random matrices has received rather little attention to the best of our knowledge coupled random matrices have been studied to some extent by mehta in this work we explain how the integrable technology can be brought to bear to gain insight into the nature of the distribution of the spectrum of coupled hermitean random matrices and the equations the associated probabilities satisfy in particular the two toda lattice its algebra of symmetries and its vertex operators will play a prominent role in this interaction namely the method is to introduce time parameters in an artificial way and to dress up a certain matrix integral with a vertex integral operator for which we find virasoro like differential equations these methods lead to very simple nonlinear third order partial differential equations for the joint statistics of the spectra of two coupled gaussian random matrices comment 56 pages published version abstract added in migration

orthogonal laurent polynomials on the unit circle extended cmv ordering and 2d toda type integrable hierarchies abstract we connect the theory of orthogonal laurent polynomials on the unit circle and the theory of toda like integrable systems using the gauss borel factorization of a cantero moral velazquez moment matrix that we construct in terms of a complex quasi definite measure supported on the unit circle the factorization of the moment matrix leads to orthogonal laurent polynomials on the unit circle and the corresponding second kind functions we obtain jacobi operators 5 term recursion relations christoffel darboux kernels and corresponding christoffel darboux formulas from this point of view in a completely algebraic way we generalize the cantero moral velazquez sequence of laurent monomials recursion relations christoffel darboux kernels and corresponding christoffel darboux formulas in this extended context we introduce continuous deformations of the moment matrix and we show how they induce a time dependent orthogonality problem related to a toda type integrable system which is connected with the well known toeplitz lattice we obtain the lax and zakharov shabat equations using the classical integrability theory tools we explicitly derive the dynamical system associated with the coefficients of the orthogonal laurent polynomials and we compare it with the classical toeplitz lattice dynamical system for the verblunsky coefficients of szeg polynomials for a positive measure discrete flows are introduced and related to darboux transformations finally we obtain the representation of the orthogonal laurent polynomials and their second kind functions using the formalism of miwa shifts in terms of functions and the subsequent bilinear equations

matrix orthogonal laurent polynomials on the unit circle and toda type integrable systems matrix orthogonal laurent polynomials in the unit circle and the theory of toda like integrable systems are connected using the gauss borel factorization of two left and a right cantero morales velazquez block moment matrices which are constructed using a quasi definite matrix measure a block gauss borel factorization problem of these moment matrices leads to two sets of biorthogonal matrix orthogonal laurent polynomials and matrix szeg polynomials which can be expressed in terms of schur complements of bordered truncations of the block moment matrix the corresponding block extension of the christoffel darboux theory is derived deformations of the quasidefinite matrix measure leading to integrable systems of toda type are studied the integrable theory is given in this matrix scenario wave and adjoint wave functions lax and zakharov shabat equations bilinear equations and discrete flows connected with darboux transformations we generalize the integrable flows of the cafasso s matrix extension of the toeplitz lattice for the verblunsky coefficients of szeg polynomials an analysis of the miwa shifts allows for the finding of interesting connections between christoffel darboux kernels and miwa shifts of the matrix orthogonal laurent polynomials

on the algebraic structure of rational discrete dynamical systems we show how singularities shape the evolution of rational discrete dynamical sys tems the stabilisation of the form of the iterates suggests a description providing among other things generalised hirota form exact evaluation of the algebraic entropy as well as remarkable polynomial factorisation properties we illustrate the phenomenon explicitly with examples covering a wide range of models

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

exponential error terms for growth functions on negatively curved surfaces in this paper we consider two counting problems associated with compact negatively curved surfaces and improve classical asymptotic estimates due to margulis in the first we show that the number of closed geodesies of length at most t has an exponential error term in the second we show that the number of geodesic arcs between two fixed points x and y of length at most t has an exponential error term the proof is based on a detailed study of the zeta function and poincar series and benefits from recent work of dolgopiat 0 introduction let v be a compact surface with negative curvature and let v be its universal cover with the lifted metric d the fundamental group n v can be realized as a group of isometries t acting on v with v t v given a point x g v9 we can introduce the orbital counting function n t defined

algebraic aspects of the discrete kp hierarchy abstract we discuss some algebraic properties of the so called discrete kp hierarchy an integrable system defined on a space of infinite matrices we give an algebraic proof of the complete integrability of the hierarchy which we achieve by means of a factorization result for infinite matrices that extends a result of m adler and p van moerbeke commun math phys 203 1999 185 207 1999 589 for the case of semi infinite moment matrices and that we call a borel decomposition

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

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

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

multivariate orthogonal polynomials and integrable systems multivariate orthogonal polynomials in d real dimensions are considered from the perspective of the cholesky factorization of a moment matrix the approach allows for the construction of corresponding multivariate orthogonal polynomials associated second kind functions jacobi type matrices and associated three term relations and also christoffel darboux formulae the multivariate orthogonal polynomials their second kind functions and the corresponding christoffel darboux kernels are shown to be quasi determinants as well as schur complements of bordered truncations of the moment matrix quasi tau functions are introduced it is proven that the second kind functions are multivariate cauchy transforms of the multivariate orthogonal polynomials discrete and continuous deformations of the measure lead to toda type integrable hierarchy being the corresponding flows described through lax and zakharov shabat equations bilinear equations are found varying size matrix nonlinear partial difference and differential equations of the 2d toda lattice type are shown to be solved by matrix coefficients of the multivariate orthogonal polynomials the discrete flows which are shown to be connected with a gauss borel factorization of the jacobi type matrices and its quasi determinants lead to expressions for the multivariate orthogonal polynomials and their second kind functions in terms of shifted quasi tau matrices which generalize to the multidimensional realm those that relate the baker and adjoint baker functions to ratios of miwa shifted tau functions in the 1d scenario in this context the multivariate extension of the elementary darboux transformation is given in terms of quasi determinants of matrices built up by the evaluation at a poised set of nodes lying in an appropriate hyperplane in r d of the multivariate orthogonal polynomials the multivariate christoffel formula for the iteration of m elementary darboux transformations is given as a quasi determinant it is shown using congruences in the space of semi infinite matrices that the discrete and continuous flows are intimately connected and determine nonlinear partial difference differential equations that involve only one site in the integrable lattice behaving as a kadomstev petviashvili type system finally a brief discussion of measures with a particular linear isometry invariance and some of its consequences for the corresponding multivariate polynomials is given in particular it is shown that the toda times that preserve the invariance condition lay in a secant variety of the veronese variety of the fixed point set of the linear isometry

a representation of the peano kernel for some quadrature rules and applications a general interpolating formula is established from this formula all newton cotes quadrature rules of the closed type can be derived some corrected interpolating polynomials are also derived and used for obtaining corresponding quadrature rules a new effective representation of the peano kernel is derived estimation of errors for these quadrature rules is established

rational maps with real multipliers let f be a rational function such that the multipliers of all repelling periodic points are real we prove that the julia set of such a function belongs to a circle combining this with a result of fatou we conclude that whenever j f belongs to a smooth curve it also belongs to a circle then we discuss rational functions whose julia sets belong to a circle

dimension rigidity in conformal structures abstract let be the limit set of a conformal dynamical system i e a kleinian group acting on either finite or infinite dimensional real hilbert space a conformal iterated function system or a rational function we give an easily expressible sufficient condition requiring that the limit set is not too much bigger than the radial limit set for the following dichotomy is either a real analytic manifold or a fractal in the sense of mandelbrot i e its hausdorff dimension is strictly greater than its topological dimension our primary focus is on the infinite dimensional case an important component of the strategy of our proof comes from the rectifiability techniques of mayer and urbanski 03 who obtained a dimension rigidity result for conformal iterated function systems including those with infinite alphabets in order to handle the infinite dimensional case both for kleinian groups and for iterated function systems we introduce the notion of pseudorectifiability a variant of rectifiability and develop a theory around this notion similar to the theory of rectifiable sets our approach also extends existing results in the finite dimensional case where it unifies the realms of kleinian groups conformal iterated function systems and rational functions for kleinian groups we improve on the rigidity result of kapovich 09 by substantially weakening its hypothesis of geometrical finiteness moreover our proof based on rectifiability is entirely different than that of kapovich which depends on homological algebra another advantage of our approach is that it allows us to use the demension of stan ko 69 as a substitute for topological dimension for example we prove that any dynamically defined version of antoine s necklace must have hausdorff dimension strictly greater than 1 i e the demension of antoine s necklace

hausdorff dimension of wild fractals we show that for every s n 2 n there exists a homogeneously embedded wild cantor set c s in r n n 3 of local hausdorff dimension s also it is shown that for every s n 2 n and for any integer k n such that 1 k s there exist everywhere wild k spheres and k cells in r n n 3 of local hausdorff dimension s

partial skew orthogonal polynomials and related integrable lattices with pfaffian tau functions skew orthogonal polynomials sops arise in the study of the n point distribution function for orthogonal and symplectic random matrix ensembles motivated by the average of characteristic polynomials of the bures random matrix ensemble studied in forrester and kieburg commun math phys 342 1 151 187 2016 we propose the concept of partial skew orthogonal polynomials psops as a modification of the sops and then the psops with a variety of special skew symmetric kernels and weight functions are addressed by considering appropriate deformations of the weight functions we derive nine integrable lattices in different dimensions as a consequence the tau functions for these systems are shown to be expressed in terms of pfaffians and the wave vectors psops in fact the tau functions also admit the multiple integral representations among these integrable lattices some of them are known while the others are novel to the best of our knowledge in particular one integrable lattice is related to the partition function of the bures ensemble besides we derive a discrete integrable lattice which can be used to compute certain vector pade approximants this yields the first example regarding the connection between integrable lattices and generalised inverse vector valued pade approximants about which hietarinta joshi and nijhoff pointed out that this field remains largely to be explored in the recent monograph hietarinta et al in discrete systems and integrability vol 54 cambridge university press cambridge 2016 section 4 4

prime number theorems and holonomies for hyperbolic rational maps for a hyperbolic rational map f of degree at least two on the riemann sphere we obtain estimates for the number of primitive periodic orbits of f ordered by their multiplier and establish equidistribution of the associated holonomies both with power saving error terms

string orthogonal polynomials string equations and 2 toda symmetries 1 the 2 toda lattice and its generic symmetries 2 a larger class of symmetries for special initial conditions 3 borel decomposition of moment matrices tau functions and string orthogonal polynomials 4 from string orthogonal polynomials to the 2 toda lattice and the string equation 5 virasoro constraints on two matrix integrals comment 45 pages latex

on decay of correlations in anosov flows there is some disagreement about the meaning of the phrase chaotic flow however there is no doubt that mixing anosov flows provides an example of such systems anosov systems were introduced and extensively studied in his classical memoir a among other things he proved the following fact known now as anosov alternative for flows either every strong stable and strong unstable manifold is everywhere dense or the flow gt is a suspension over an anosov diffeomorphism by a constant roof function if the first alternative holds gt is mixing with respect to every gibbs measure see pp2 now the natural question is what is the estimated rate of mixing this is certainly one of the simplest questions concerning correlation decay in continuous time systems nevertheless the only results obtained until recently dealt with the case when the system discussed had an additional algebraic structure the easier case of anosov diffeomorphisms can be treated by the methods of thermodynamic formalism of sinai ruelle and bowen b2 namely one uses markov partitions to construct an isomorphism between the diffeomorphism and a subshift of finite type and then proves that all such subshifts are exponentially mixing this method would succeed also for flows if any suspension over a subshift of finite type had exponentially decaying correlations however the simplest example suspensions with locally constant roof functions never have such a property ri one can use the above observation to produce examples of axiom a flows with arbitrary slow correlation decay it became clear therefore that some additional geometric properties should be taken into account in recent work chernov chl ch2 has employed a uniform nonintegrability condition to get a subexponential estimate for correlation functions for geodesic flows on surfaces of variable negative curvature his method relies on the technique of markov approximation developed in chl the aim of this paper is to combine geometric considerations of chernov with the thermodynamic formalism approach the later method seems to be more appropriate than markov approximations since it gives simpler

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

some new error inequalities for a generalized quadrature rule of open type some new error inequalities for a generalized quadrature rule of open type are established especially two sharp inequalities are derived when n is an odd and an even integer respectively

the thermodynamic formalism for expanding maps letf x x be an expanding map of a compact space small distances are increased by a factor 1 a generating function z is defined which countsf periodic points with a weight one can express in terms of nonstandard fredholm determinants of certain transfer operators which can be studied by methods borrowed from statistical mechanics in this paper we review the spectral properties of the transfer operators and the corresponding analytic properties of z gibbs distributions and applications to julia sets are also discussed some new results are proved and some natural conjectures are proposed

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 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

calogero moser hierarchy and kp hierarchy identification of the hierarchy of calogero moser dynamical systems with the dynamical systems of poles of certain solutions to the kp hierarchy is achieved

dimensions and measures in infinite iterated function systems the hausdorff and packing measures and dimensions of the limit sets of iterated function systems generated by countable families of conformal contractions are investigated conformal measures for such systems reflecting geometric properties of the limit set are introduced proved to exist and to be unique the existence of a unique invariant probability equivalent to the conformal measure is derived our methods employ the concepts of the perron frobenius operator symbolic dynamics on a shift space with an infinite alphabet and the properties of the above mentioned ergodic invariant measure a formula for the hausdorff dimension of the limit set in terms of the pressure function is derived fractal phenomena not exhibited by finite systems are shown to appear in the infinite case in particular a variety of conditions are provided for hausdorff and packing measures to be positive or finite and a number of examples are described showing the appearance of various possible combinations for these quantities one example given special attention is the limit set associated to the complex continued fraction expansion in particular lower and upper estimates for its hausdorff dimension are given a large natural class of systems whose limit sets are dimensionless in the restricted sense is described

vertex operator solutions to the discrete kp hierarchy vertex operators which are disguised darboux maps transform solutions of the kp equation into new ones in this paper we show that the bi infinite sequence obtained by darboux transforming an arbitrary kp solution recursively forward and backwards yields a solution to the discrete kp hierarchy the latter is a kp hierarchy where the continuous space x variable gets replaced by a discrete n variable the fact that these sequences satisfy the discrete kp hierarchy is tantamount to certain bilinear relations connecting the consecutive kp solutions in the sequence at the grassmannian level these relations are equivalent to a very simple fact which is the nesting of the associated infinite dimensional planes flag the discrete kp hierarchy can thus be viewed as a container for an entire ensemble of vertex or darboux generated kp solutions it turns out that many new and old systems lead to such discrete semi infinite solutions like sequences of soliton solutions with more and more solitons sequences of calogero moser systems having more and more particles just to mention a few examples this is developed in 3 in this paper as another example we show that the q kp hierarchy maps via a kind of fourier transform into the discrete kp hierarchy enabling us to write down a very large class of solutions to the q kp hierarchy this was also reported in a brief note 4

a unified generalization of some quadrature rules and error bounds by introducing a parameter we give a unified generalization of some quadrature rules which not only unify the recent results about error bounds for generalized mid point trapezoid and simpson s rules but also give some new error bounds for other quadrature rules as special cases especially two sharp error inequalities are derived when n is an odd and an even integer respectively

eigenvalue density in hermitian matrix models by the lax pair method in this paper a new method is discussed to derive the eigenvalue density in a hermitian matrix model with a general potential the density is considered on one interval or multiple disjoint intervals the method is based on lax pair theory and the cayley hamilton theorem by studying the orthogonal polynomials associated with the hermitian matrix model it is obtained that the restriction conditions for the parameters in the density are connected to the discrete painleve i equation and the results are related to the scalar riemann hilbert problem some special density functions are also discussed in association with the known results in this subject

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

divergence groups have the bowen property we show that every deformation of a divergence type fuchsian group has a limit set which is either a circle or has dimension strictly larger than 1 this is known to be false for all convergence type groups and hence solves bowen s problem in the general case the proof uses a theorem of dennis sullivan s about convex hulls in hyperbolic 3 space and we give a new simpler proof of this result as well