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

finitized conformal spectrum of the ising model on the cylinder and torus abstract the spectrum of the critical ising model on a lattice with cylindrical and toroidal boundary conditions is calculated by commuting transfer matrix methods using a simple truncation procedure we obtain the natural finitizations of the conformal spectra recently proposed by melzer these finitizations imply polynomial identities which in the large lattice limit give rise to the rogers ramanujan identities for the c 1 2 virasoro characters

singular value correlation functions for products of wishart random matrices we consider the product of m quadratic random matrices with complex elements and no further symmetry where all matrix elements of each factor have a gaussian distribution this generalizes the classical wishart laguerre gaussian unitary ensemble with m 1 in this paper we first compute the joint probability distribution for the singular values of the product matrix when the matrix size n and the number m are fixed but arbitrary this leads to a determinantal point process which can be realized in two different ways first it can be written as a one matrix singular value model with a nonstandard jacobian or second for m 2 as a two matrix singular value model with a set of auxiliary singular values and a weight proportional to the meijer g function for both formulations we determine all singular value correlation functions in terms of the kernels of biorthogonal polynomials which we explicitly construct they are given in terms of the hypergeometric and meijer g functions generalizing the laguerre polynomials for m 1 our investigation was motivated from applications in telecommunication of multi layered scattering multiple input and multiple output channels we present the ergodic mutual information for finite n for such a channel model with m 1 layers of scatterers as an example

products of rectangular random matrices singular values and progressive scattering we discuss the product of m rectangular random matrices with independent gaussian entries which have several applications including wireless telecommunication and econophysics for complex matrices an explicit expression for the joint probability density function is obtained using the harish chandra itzykson zuber integration formula explicit expressions for all correlation functions and moments for finite matrix sizes are obtained using a two matrix model and the method of biorthogonal polynomials this generalizes the classical result for the so called wishart laguerre gaussian unitary ensemble or chiral unitary ensemble at m 1 and previous results for the product of square matrices the correlation functions are given by a determinantal point process where the kernel can be expressed in terms of meijer g functions we compare the results with numerical simulations and known results for the macroscopic level density in the limit of large matrices the location of the end points of support for the latter are analyzed in detail for general m finally we consider the so called ergodic mutual information which gives an upper bound for the spectral efficiency of a mimo communication channel with multifold scattering

finitized conformal spectra of the ising model on the klein bottle and mobius strip we study the conformal spectra of the critical square lattice ising model on the klein bottle and mobius strip using yang baxter techniques and the solution of functional equations in particular we obtain expressions for the finitized conformal partition functions in terms of finitized virasoro characters this demonstrates that yang baxter techniques and functional equations can be used to study the conformal spectra of more general exactly solvable lattice models in these topologies the results rely on certain properties of the eigenvalues which are confirmed numerically

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

ising model on nonorientable surfaces exact solution for the mobius strip and the klein bottle closed form expressions are obtained for the partition function of the ising model on an mxn simple quartic lattice embedded on a mobius strip and a klein bottle the solutions all lead to the same bulk free energy but for finite m and n the expressions are different depending on whether the strip width m is odd or even finite size corrections at criticality are analyzed and compared with those under cylindrical and toroidal boundary conditions our results are consistent with the conformal field prediction of a central charge c 1 2 provided that the twisted mobius boundary condition is regarded as a free or fixed boundary

integral points close to a space curve we establish sharp lower and upper bounds for the number of integral points near dilations of a space curve with nowhere vanishing torsion

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

on the asymptotics of dimers on tori we study asymptotics of the dimer model on large toric graphs let mathbb l be a weighted mathbb z 2 periodic planar graph and let mathbb z 2 e be a large index sublattice of mathbb z 2 for mathbb l bipartite we show that the dimer partition function textsf textit z e on the quotient mathbb l mathbb z 2 e has the asymptotic expansion begin aligned textsf textit z exp a mathbf f 0 textsf fsc o 1 end aligned where a is the area of mathbb l mathbb z 2 e mathbf f 0 is the free energy density in the bulk and textsf fsc is a finite size correction term depending only on the conformal shape of the domain together with some parity type information assuming a conjectural condition on the zero locus of the dimer characteristic polynomial we show that an analogous expansion holds for mathbb l non bipartite the functional form of the finite size correction differs between the two classes but is universal within each class our calculations yield new information concerning the distribution of the number of loops winding around the torus in the associated double dimer models

universal entropy of conformal critical theories on a klein bottle we show that rational conformal field theories in 1 1 dimensions on a klein bottle with a length l and width satisfying l have a universal entropy this universal entropy depends on the quantum dimensions of the primary fields and can be accurately extracted by taking a proper ratio between the klein bottle and torus partition functions enabling the characterization of conformal critical theories the result is checked against exact calculations in quantum spin 1 2 xy and ising chains

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

the supercritical regime in the normal matrix model with cubic potential abstract the normal matrix model with a cubic potential is ill defined and it develops a critical behavior in finite time we follow the approach of bleher and kuijlaars to reformulate the model in terms of orthogonal polynomials with respect to a hermitian form this reformulation was shown to capture the essential features of the normal matrix model in the subcritical regime namely that the zeros of the polynomials tend to a number of segments the motherbody inside a domain the droplet that attracts the eigenvalues in the normal matrix model in the present paper we analyze the supercritical regime and we find that the large n behavior is described by the evolution of a spectral curve satisfying the boutroux condition the boutroux condition determines a system of contours 1 consisting of the motherbody and whiskers sticking out of the domain we find a second critical behavior at which the original motherbody shrinks to a point at the origin and only the whiskers remain in the regime before the second criticality we also give strong asymptotics of the orthogonal polynomials by means of a steepest descent analysis of a 3 3 matrix valued riemann hilbert problem it follows that the zeros of the orthogonal polynomials tend to 1 with the exception of at most three spurious zeros

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

rational points near planar curves and diophantine approximation abstract in this paper we establish asymptotic formulae with optimal errors for the number of rational points that are close to a planar curve which unify and extend the results of beresnevich dickinson velani 6 and vaughan velani 22 furthermore we complete the lebesgue theory of diophantine approximation on weakly non degenerate planar curves that was initially developed by beresnevich zorin 5 in the divergence case

fa yueh wu s contributions in physics abstract we sketch a subset of professor f y wu s contributions in lattice statistical mechanics solid state physics graph theory enumerative combinatorics and other domains of physics and mathematics we will recall some of f y wu s most important and well known classic results and we will also sketch some of his most recent researches dedicated to the connections of lattice statistical mechanical models with deep problems in pure mathematics

the density of rational points near hypersurfaces we establish a sharp asymptotic formula for the number of rational points up to a given height and within a given distance from a hypersurface our main innovation is a bootstrap method that relies on the synthesis of poisson summation projective duality and the method of stationary phase this has surprising applications to counting rational points lying on the manifold indeed we are able to prove an analogue of serre s dimension growth conjecture originally stated for projective varieties in this general setup as another consequence of our main counting result we solve the generalized baker schmidt problem in the simultaneous setting for hypersurfaces

explicit bounds for rational points near planar curves and metric diophantine approximation abstract the primary goal of this paper is to complete the theory of metric diophantine approximation initially developed in beresnevich et al 2007 10 for c 3 non degenerate planar curves with this goal in mind here for the first time we obtain fully explicit bounds for the number of rational points near planar curves further introducing a perturbational approach we bring the smoothness condition imposed on the curves down to c 1 lowest possible this way we broaden the notion of non degeneracy in a natural direction and introduce a new topologically complete class of planar curves to the theory of diophantine approximation in summary our findings improve and complete the main theorems of beresnevich et al 2007 10 and extend the celebrated theorem of kleinbock and margulis 1998 20 in dimension 2 beyond the notion of non degeneracy

singular values of products of ginibre random matrices multiple orthogonal polynomials and hard edge scaling limits akemann ipsen and kieburg recently showed that the squared singular values of products of m rectangular random matrices with independent complex gaussian entries are distributed according to a determinantal point process with a correlation kernel that can be expressed in terms of meijer g functions we show that this point process can be interpreted as a multiple orthogonal polynomial ensemble we give integral representations for the relevant multiple orthogonal polynomials and a new double contour integral for the correlation kernel which allows us to find its scaling limits at the origin hard edge the limiting kernels generalize the classical bessel kernels for m 2 they coincide with the scaling limits found by bertola gekhtman and szmigielski in the cauchy laguerre two matrix model which indicates that these kernels represent a new universality class in random matrix theory

geometric postulation of a smooth function and the number of rational points this paper is devoted to giving refinements and extensions of some of the results of bombieri and the author 1 obtaining upper bounds for the number of integral lattice points on the graphs of functions consider a sufficiently smooth function f x with graph and a positive integer d the main device of that paper was to consider integral points on that do not lie on any real algebraic curve of degree d the main lemma of 1 shows that such points cannot be too close together relative to certain norms of the function we pursue here two different goals relative to the main lemma the first is to obtain local conditions on the function f x that control the multiplicity of the intersection of with any algebraic curve of degree d this is essentially an investigation into the hypotheses of the main lemma and constitutes the geometric postulation of the title indeed in sections 2 and 3 we will obtain such conditions for any linear space of real algebraic curves the applications to integral points are given in section 5 for example we show that if f x c on 0 1 and w f 2 f f 3f 0 f iv 4f 6f f 5f iv 20f is nowhere zero then for every e 0

partition functions and finite size scalings of ising model on helical tori the exact closed forms of the partition functions of a two dimensional ising model on square lattices with twisted boundary conditions are given the constructions of helical tori are unambiguously related to the twisted boundary conditions by virtue of the sl 2 z transforms the numerical analyses on the deviations of the specific heat peaks away from the bulk critical temperature reveal that the finite size effect of herical tori is independent of the chirality

exact finite size scaling corrections to the critical two dimensional ising model on a torus ii triangular and hexagonal lattices we compute the finite size corrections to the free energy internal energy and specific heat of the critical two dimensional spin 1 2 ising model on triangular and hexagonal lattices wrapped on a torus we find the general form of the finite size corrections to these quantities as well as explicit formulae for the first coefficients of each expansion we analyse the implications of these findings for the renormalization group description of the model

asymptotic expansion for the resistance between two maximally separated nodes on an m by n resistor network we analyze the exact formulae for the resistance between two arbitrary notes in a rectangular network of resistors under free periodic and cylindrical boundary conditions obtained by wu j phys a 37 6653 2004 based on such expression we then apply the algorithm of ivashkevich izmailian and hu j phys a 35 5543 2002 to derive the exact asymptotic expansions of the

partition function zeroes of a self dual ising model abstract we consider the ising model on an m n rectangular lattice with an asymmetric self dual boundary condition and derive a closed form expression for its partition function we show that zeroes of the partition function are given by the roots of a polynomial equation of degree 2m 1 which trace out certain loci in the complex temperature plane particularly it is shown that a real solutions of the polynomial equations always lead to zeroes on the unit circle and a segment of the negative real axis and b all temperature zeroes lie on two circles in the limit of m for any n closed form expressions of the loci as well as the density of zero distributions in the limit of n are derived for m 1 and 2 in addition we explain the reason of and establish the criterion for partition function zeroes of any self dual spin model to reside precisely on the unit circle this elucidates a recent finding in the case of the self dual potts model

self similarity in the classification of finite size scaling functions for toroidal boundary conditions the conventional periodic boundary conditions in two dimensions are extended to general boundary conditions prescribed by primitive vector pairs that may not coincide with the coordinate axes this extension is shown to be unambiguously specified by the twisting scheme equivalent relations between different twist settings are constructed explicitly the classification of finite size scaling functions is discussed based on the equivalent relations a self similar pattern for distinct classes of finite size scaling functions is shown to appear on the plane that parametrizes the toroidal geometry

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

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

winding clusters in percolation on the torus and the mobius strip using a simulation technique introduced recently we study winding clusters in percolation on the torus and the mobius strip for different aspect ratios the asynchronous parallelization of the simulation makes very large system and sample sizes possible our high accuracy results are fully consistent with predictions from conformal field theory the numerical results for the mobius strip and the number distribution of winding clusters on the torus await theoretical explanation to our knowledge this study is the first of its kind

non intersecting squared bessel paths and multiple orthogonal polynomials for modified bessel weights we study a model of n non intersecting squared bessel processes in the confluent case all paths start at time t 0 at the same positive value x a remain positive and are conditioned to end at time t t at x 0 in the limit n after appropriate rescaling the paths fill out a region in the tx plane that we describe explicitly in particular the paths initially stay away from the hard edge at x 0 but at a certain critical time t the smallest paths hit the hard edge and from then on are stuck to it for t t we obtain the usual scaling limits from random matrix theory namely the sine airy and bessel kernels a key fact is that the positions of the paths at any time t constitute a multiple orthogonal polynomial ensemble corresponding to a system of two modified bessel type weights as a consequence there is a 3 3 matrix valued riemann hilbert problem characterizing this model that we analyze in the large n limit using the deift zhou steepest descent method there are some novel ingredients in the riemann hilbert analysis that are of independent interest

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

the number of integral points on arcs and ovals integral lattice points and that the exponent and constant are best possible however swinnerton dyer 10 showed that the preceding result can be substantially improved if we start with a fixed c strictly convex arc and consider the number of lattice points on t the dilation of by a factor t t 1 this of course is the same as asking for rational points mn n n on as n in fact swinnerton dyer proves a bound of type t zz c e t 3 5 e

large n limit of gaussian random matrices with external source part iii double scaling limit we consider the double scaling limit in the random matrix ensemble with an external source 1 over z n e n hbox tr 1 over 2 m 2 am dm defined on n n hermitian matrices where a is a diagonal matrix with two eigenvalues a of equal multiplicities the value a 1 is critical since the eigenvalues of m accumulate as n on two intervals for a 1 and on one interval for 0 a 1 these two cases were treated in parts i and ii where we showed that the local eigenvalue correlations have the universal limiting behavior known from unitary random matrix ensembles for the critical case a 1 new limiting behavior occurs which is described in terms of pearcey integrals as shown by brezin and hikami and tracy and widom we establish this result by applying the deift zhou steepest descent method to a 3 3 matrix valued riemann hilbert problem which involves the construction of a local parametrix out of pearcey integrals we resolve the main technical issue of matching the local pearcey parametrix with a global outside parametrix by modifying an underlying riemann surface

the density of rational points on curves and surfaces let c be an irreducible projective curve of degree d in mathbb p 3 defined over overline mathbb q it is shown that c has o varepsilon d b 2 d varepsilon rational points of height at most b for any varepsilon 0 uniformly for all curves c this result extends an estimate of bombieri and pila duke math j 59 1989 337 357 to projective curves for a projective surface s in mathbb p 3 of degree d ge 3 it is shown that there are o varepsilon d b 2 varepsilon rational points of height at most b of which at most o varepsilon d b 52 27 varepsilon do not lie on a rational line in s for non singular surfaces one may reduce the exponent to 4 3 16 9d for d 4 or 5 or max 1 3 sqrt d 2 d 1 for d ge 6 even for the surface x 1 d x 2 d x 3 d x 4 d this last result improves on the previous best known as a further application it is shown that almost all integers represented by an irreducible binary form f x y in mathbb z x y have essentially only one such representation this extends a result of hooley j reine angew math 226 1967 30 87 which concerned cubic forms only the results are not restricted to projective surfaces and as an application of other results in the paper it is shown that x 1 x 2 x 3 in mathbb n 3 x 1 d x 2 d x 3 d n ll varepsilon d n theta d varepsilon with theta frac 2 sqrt d frac 2 d 1 when d ge 8 this provides the first non trivial bound for the number of representations as a sum of three d th powers

probability distribution of magnetization in the one dimensional ising model effects of boundary conditions finite size scaling functions are investigated both for the mean square magnetization fluctuations and for the probability distribution of the magnetization in the one dimensional ising model the scaling functions are evaluated in the limit of the temperature going to zero t 0 the size of the system going to infinity n while n 1 tanh j kbt is kept finite j being the nearest neighbour coupling exact calculations using various boundary conditions periodic antiperiodic free block demonstrate explicitly how the scaling functions depend on the boundary conditions we also show that the block small part of a large system magnetization distribution results are identical to those obtained for free boundary conditions

diophantine approximation on planar curves and the distribution of rational points let c be a nondegenerate planar curve and for a real positive decreasing function let c denote the set of simultaneously approximable points lying on c we show that c is of khintchine type for divergence i e if a certain sum diverges then the one dimensional lebesgue measure on c of c is full we also obtain the hausdorff measure analogue of the divergent khintchine type result in the case that c is a rational quadric the convergence counterparts of the divergent results are also obtained furthermore for functions with lower order in a critical range we determine a general exact formula for the hausdorff dimension of c these results constitute the first precise and general results in the theory of simultaneous diophantine approximation on manifolds

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

exact partition functions of the ising model on m n planar lattices with periodic aperiodic boundary conditions the grassmann path integral approach is used to calculate exact partition functions of the ising model on m n square sq plane triangular pt and honeycomb hc lattices with periodic periodic pp periodic antiperiodic pa antiperiodic periodic ap and antiperiodic antiperiodic aa boundary conditions the partition functions are used to calculate and plot the specific heat c kb as a function of temperature kbt j we find that for the n n sq lattice c kb for pa and ap boundary conditions are different from those for aa boundary conditions but for the n n pt and hc lattices c kb for ap pa and aa boundary conditions have the same values our exact partition functions might also be useful for understanding the effects of lattice structures and boundary conditions on critical finite size corrections of the ising model

exact finite size scaling corrections to the critical two dimensional ising model on a torus we analyse the finite size corrections to the energy and specific heat of the critical two dimensional spin ising model on a torus we extend the analysis of ferdinand and fisher to compute the correction of order l 3 to the energy and the corrections of order l 2 and l 3 to the specific heat we also obtain general results on the form of the finite size corrections to these quantities only integer powers of l 1 occur unmodified by logarithms except of course for the leading log l term in the specific heat and the energy expansion contains only odd powers of l 1 in the specific heat expansion any power of l 1 can appear but the coefficients of the odd powers are proportional to the corresponding coefficients of the energy expansion

universality in the two matrix model a riemann hilbert steepest descent analysis the eigenvalue statistics of a pair m 1 m 2 of n x n hermitian matrices taken randomly with respect to the measure 1 z n exp n tr v m 1 w m 2 tau m 1 m 2 dm 1 dm 2 can be descri

large n limit of gaussian random matrices with external source part i we continue the study of the hermitian random matrix ensemble with external source open image in new window where a has two distinct eigenvalues a of equal multiplicity this model exhibits a phase transition for the value a 1 since the eigenvalues of m accumulate on two intervals for a 1 and on one interval for 0 1 was treated in part i where it was proved that local eigenvalue correlations have the universal limiting behavior which is known for unitarily invariant random matrices that is limiting eigenvalue correlations are expressed in terms of the sine kernel in the bulk of the spectrum and in terms of the airy kernel at the edge in this paper we establish the same results for the case 0 a 1 as in part i we apply the deift zhou steepest descent analysis to a 3 3 matrix riemann hilbert problem due to the different structure of an underlying riemann surface the analysis includes an additional step involving a global opening of lenses which is a new phenomenon in the steepest descent analysis of riemann hilbert problems

eigenvalue statistics for product complex wishart matrices the eigenvalue statistics for complex wishart matrices where is equal to the product of r complex gaussian matrices and the inverse of s complex gaussian matrices are considered in the case r s the exact form of the global density is computed the averaged characteristic polynomial for the corresponding generalized eigenvalue problem is calculated in terms of a particular generalized hypergeometric function for finite n the eigenvalue probability density function is computed and is shown to be an example of a biorthogonal ensemble a double contour integral form of the corresponding correlation kernel is derived which allows the hard edge scaled limit to be computed the limiting kernel is given in terms of certain meijer g functions and is identical to that found in the recent work of kuijlaars and zhang in the case s 0 properties of the kernel and corresponding correlation functions are discussed

continuous time quantum walks on nonorientable surfaces analytical solutions for mobius strips and klein bottles in this paper we present an extensive analytical study of continuous time quantum walks ctqws occurring on two dimensional lattices under various boundary conditions focusing on the m n lattice wrapped on mobius strips and klein bottles which are featured by the twisted boundary conditions we find that the eigenvalue spectra and transport efficiency both quantum mechanical and classical for the two structures show comparable results with those for other well known two dimensional lattices including rectangles cylinders and tori we also demonstrate that the behaviors of ctqws depend on the initial node of the excitation and the network size both on short and long timescales in addition we discover the asymmetric behaviors of limiting probabilities for mobius strips and klein bottles which are quite different from each other and are also compared to those discovered in other two dimensional networks our work provides a comprehensive understanding of recent results about ctqws on two dimensional lattices and sheds light on quantum dynamics on lattices especially those with different boundary conditions

dropping the independence singular values for products of two coupled random matrices we study the singular values of the product of two coupled rectangular random matrices as a determinantal point process each of the two factors is given by a parameter dependent linear combination of two independent complex gaussian random matrices which is equivalent to a coupling of the two factors via an itzykson zuber term we prove that the squared singular values of such a product form a biorthogonal ensemble and establish its exact solvability the parameter dependence allows us to interpolate between the singular value statistics of the laguerre ensemble and that of the product of two independent complex ginibre ensembles which are both known we give exact formulae for the correlation kernel in terms of a complex double contour integral suitable for the subsequent asymptotic analysis in particular we derive a christoffel darboux type formula for the correlation kernel based on a five term recurrence relation for our biorthogonal functions it enables us to find its scaling limit at the origin representing a hard edge the resulting limiting kernel coincides with the universal meijer g kernel found by several authors in different ensembles we show that the central limit theorem holds for the linear statistics of the singular values and give the limiting variance explicitly

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

diophantine approximation on planar curves the convergence theory the convergence theory for the set of simultaneously approximable points lying on a planar curve is established our results complement the divergence theory developed in 1 and thereby completes the general metric theory for planar curves

universality conjecture and results for a model of several coupled positive definite matrices the paper contains two main parts in the first part we analyze the general case of p geq 2 matrices coupled in a chain subject to cauchy interaction similarly to the itzykson zuber interaction model the eigenvalues of the cauchy chain form a multi level determinantal point process we first compute all correlations functions in terms of cauchy biorthogonal polynomials and locate them as specific entries of a p 1 times p 1 matrix valued solution of a riemann hilbert problem in the second part we fix the external potentials as classical laguerre weights we then derive strong asymptotics for the cauchy biorthogonal polynomials when the support of the equilibrium measures contains the origin as a result we obtain a new family of universality classes for multi level random determinantal point fields which include the bessel nu universality for 1 level and the meijer g universality for 2 level our analysis uses the deift zhou nonlinear steepest descent method and the explicit construction of a p 1 times p 1 origin parametrix in terms of meijer g functions the solution of the full riemann hilbert problem is derived rigorously only for p 3 but the general framework of the proof can be extended to the cauchy chain of arbitrary length p

rational points near manifolds and metric diophantine approximation this work is motivated by problems on simultaneous diophantine approximation on manifolds namely establishing khintchine and jarn k type theorems for submanifolds of r n these problems have attracted a lot of interest since kleinbock and margulis proved a related conjecture of alan baker and v g sprind zuk they have been settled for planar curves but remain open in higher dimensions in this paper khintchine and jarn k type divergence theorems are established for arbitrary analytic nondegenerate manifolds regardless of their dimension the key to establishing these results is the study of the distribution of rational points near manifolds a very attractive topic in its own right here for the rst time we obtain sharp lower bounds for the number of rational points near nondegenerate manifolds in dimensions n 2 and show that they are ubiquitous that is uniformly distributed

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

eigenvalues and singular values of products of rectangular gaussian random matrices we derive exact analytic expressions for the distributions of eigenvalues and singular values for the product of an arbitrary number of independent rectangular gaussian random matrices in the limit of large matrix dimensions we show that they both have power law behavior at zero and determine the corresponding powers we also propose a heuristic form of finite size corrections to these expressions which very well approximates the distributions for matrices of finite dimensions

cauchy laguerre two matrix model and the meijer g random point field we apply the general theory of cauchy biorthogonal polynomials developed in bertola et al commun math phys 287 3 983 1014 2009 and bertola et al j approx th 162 4 832 867 2010 to the case associated with laguerre measures in particular we obtain explicit formulae in terms of meijer g functions for all key objects relevant to the study of the corresponding biorthogonal polynomials and the cauchy two matrix model associated with them the central theorem we prove is that a scaling limit of the correlation functions for eigenvalues near the origin exists and is given by a new determinantal two level random point field the meijer g random field we conjecture that this random point field leads to a novel universality class of random fields parametrized by exponents of laguerre weights we express the joint distributions of the smallest eigenvalues in terms of suitable fredholm determinants and evaluate them numerically we also show that in a suitable limit the meijer g random field converges to the bessel random field and hence the behavior of the eigenvalues of one of the two matrices converges to the one of the laguerre ensemble

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