second and fourth painleve hierarchies and jimbo miwa linear problems the relations between the different linear problems for painleve equations is an intriguing open problem here we consider our previously given second and fourth painleve hierarchies publ res inst math sci kyoto 37 327 347 2001 and show that they could alternatively have been derived using the linear problems of jimbo and miwa that is we give a gauge transformation of our linear problems for these two hierarchies which maps those of the second and fourth painleve equations themselves onto those of jimbo and miwa

determinants of laplacians the determinant of the laplacian on spinor fields on a riemann surface is evaluated in terms of the value of the selberg zeta function at the middle of the critical strip a key role in deriving this relation is played by the barnes double gamma function

on the τ function of the painleve equations abstract concerning the hamiltonian structure associated with painleve equations we define the function related to a hamiltonian and show that the correlation function given in the study of the two dimensional ising model is a function for the third painleve equation the list of the hamiltonians which are polynomials in the two canonical variables is given and we prove that the function related to each of them is holomorphic

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

finite n fluctuation formulas for random matrices for the gaussian and laguerre random matrix ensembles the probability density function p d f for the linear statistic jn 1 xj x is computed exactly and shown to satisfy a central limit theorem asn for the circular random matrix ensemble the p d f s for the statistics jn 1 j and jn 1 log 2 sin j 2 are calculated exactly by using a constant term identity from the theory of the selberg integral and are also shown to satisfy a central limit theorem asn

confluent kz equations for mathfrak sl _n with poincare rank 2 at infinity we construct confluent kz equations with poincar rank 2 at infinity for the case of and the integral representation for the solutions hamiltonians of these confluent kz equations are derived from suitable quantization of dlog constructed in the theory of monodromy preserving deformation mpd in jimbo et al 1981 physica d 2 306 52 our confluent kz equations can be viewed as a quantization of mpd with poincar rank 2 at infinity

on a new non isospectral variant of the boussinesq hierarchy we give a new non isospectral extension to 2 1 dimensions of the boussinesq hierarchy such a non isospectral extension of the third order scattering problem xxx u x v 0 has not been considered previously this extends our previous results on one component hierarchies in 2 1 dimensions associated to third order non isospectral scattering problems we characterize our entire 2 1 dimensional hierarchy and its linear problem using a single partial differential equation and its corresponding non isospectral scattering problem this then allows an alternative approach to the construction of linear problems for the entire 2 1 dimensional hierarchy reductions of this hierarchy yield new integrable hierarchies of systems of ordinary differential equations together with their underlying linear problems in particular we obtain a fourth painleve hierarchy i e a hierarchy of ordinary differential equations having the fourth painleveequation as its first member we also obtain a hierarchy having as its first member a generalization of an equation defining a new transcendent due to cosgrove

the kontsevich matrix integral convergence to the painleve hierarchy and stokes phenomenon we show that the kontsevich integral on n times n matrices n infty is the isomonodromic tau function associated to a 2 times 2 riemann hilbert problem the approach allows us to gain control of the analysis of the convergence as n to infty by an appropriate choice of the external source matrix in kontsevich s integral we show that the limit produces the isomonodromic tau function of a special tronquee solution of the first painleve hierarchy and we identify the solution in terms of the stokes data of the associated linear problem we also show that there are several tau functions that are analytic in appropriate sectors of the space of parameters and that the formal witten kontsevich tau function is the asymptotic expansion of each of them in their respective sectors thus providing an analytic tool to analyze its nonlinear stokes phenomenon

antiquantization isomonodromy and integrability an extended analysis of links between linear differential equations and the nonlinear painleve equation pv i is given for linear equations second order equations in different forms as well as various first order systems are chosen the role of an accessory parameter is explained the relationship to the schlesinger system is made clear an extended analysis of links between linear differential equations and the nonlinear painleve equation pv i is given for linear equations second order equations in different forms as well as various first order systems are chosen the role of an accessory parameter is explained the relationship to the schlesinger system is made clear

string correlation functions of the spin 1 2 heisenberg xxz chain we calculate certain string correlation functions originally introduced as order parameters in integer spin chains for the spin 1 2 xxz heisenberg chain at zero temperature and in the thermodynamic limit for small distances we obtain exact results from bethe ansatz and exact diagonalization whereas in the large distance limit field theoretical arguments yield an asymptotic algebraic decay we also make contact with two point spin correlation functions in the asymptotic limit

canonical structure and symmetries of the schlesinger equations the schlesinger equations s n m describe monodromy preserving deformations of order m fuchsian systems with n 1 poles they can be considered as a family of commuting time dependent hamiltonian systems on the direct product of n copies of m m matrix algebras equipped with the standard linear poisson bracket in this paper we present a new canonical hamiltonian formulation of the general schlesinger equations s n m for all n m and we compute the action of the symmetries of the schlesinger equations in these coordinates

mean behaviour of multiplicative functions on additive arithmetical semigroups in this paper the authors prove chebyshev type estimates and asymptotic formulae for the prime elements in general additive arithmetical semigroups mean value theorems with remainder term estimates are proved for a large class of multiplicative functions on additive arithmetical semigroups as applications the authors derive asymptotic formulae for the mean behavior of prime divisor functions and of distinct degrees of prime factors the paper continues the investigations of the authors paper l

a note on the lax pairs for painleveequations for the classical painleveequations besides the method of similarity reduction of lax pairs for integrable partial differential equations two ways are known for lax pair generation the first is based on the confluence procedure in fuchs linear ode with four regular singularities isomonodromy deformation which is governed by the sixth painleveequation the second method treats the hypergeometric equation and confluent hypergeometric equations as the isomonodromy deformation equations for the triangular systems of odes in whose non triangular extensions give rise to the lax pairs for the painleveequations the theory of integrable integral operators suggests a new way of lax pair generation for the classical painleveequations this method involves a special kind of gauge transformation that is applied to linear systems which are exactly solvable in terms of the classical special functions some of the lax pairs we introduce are known others are new the question of gauge equivalence of different lax pairs for the painleveequations is considered as well

some identities for determinants of structured matrices abstract in this paper we establish several relations between the determinants of the following structured matrices hankel matrices symmetric toeplitz hankel matrices and toeplitz matrices using known results for the asymptotic behavior of toeplitz determinants these identities are used in order to obtain fisher hartwig type results on the asymptotics of certain skew symmetric toeplitz determinants and certain hankel determinants

extreme superposition rogue waves of infinite order and the painleve iii hierarchy we study the fundamental rogue wave solutions of the focusing nonlinear schrodinger equation in the limit of large order using a recently proposed riemann hilbert representation of the rogue wave solution of arbitrary order k we establish the existence of a limiting profile of the rogue wave in the large k limit when the solution is viewed in appropriate rescaled variables capturing the near field region where the solution has the largest amplitude the limiting profile is a new particular solution of the focusing nonlinear schrodinger equation in the rescaled variables the rogue wave of infinite order which also satisfies ordinary differential equations with respect to space and time the spatial differential equations are identified with certain members of the painleve iii hierarchy we compute the far field asymptotic behavior of the near field limit solution and compare the asymptotic formulas with the exact solution using numerical methods for solving riemann hilbert problems in a certain transitional region for the asymptotics the near field limit function is described by a specific globally defined tritronquee solution of the painleve ii equation these properties lead us to regard the rogue wave of infinite order as a new special function

on the solvability of the discrete second painleve equation the inverse monodromy method for studying the riemann hilbert problem associated with classical painlev e equations is applied to the discrete second painlev e equation

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

the onsager formula the fisher hartwig conjecture and their influence on research into toeplitz operators this is not primarily a paper about applications of mathematics to statistical physics but rather a report on how a particular problem of statistical physics has resulted in an extensive mathematical theory the problem alluded to is the computation of the spontaneous magnetizationm o t of the two dimensional ising model with nearest neighbor interactions whose solution for temperaturest below the curie pointt c was given by the famous formula of lars onsager in 1948 the theory grown out of this formula is the edifice of toeplitz determinants matrices and operators

the fisher hartwig conjecture and toeplitz eigenvalues abstract the conjecture of fisher and hartwig published in 1968 describes the asymptotic expansion of toeplitz determinants with singular generating functions for more than twenty years progress was made in extending the validity of the conjecture but recent computer experiments led to counterexamples that show the limits of the original conjecture and pointed the way to a revised conjecture this paper describes the history of the problem several numerical examples and the revised conjecture

model ecosystems with random nonlinear interspecies interactions the principle of competitive exclusion in ecology establishes that two species living together cannot occupy the same ecological niche here we present a model ecosystem in which the species are described by a series of phenotypic characters and the strength of the competition between two species is given by a nondecreasing modulating function of the number of common characters using analytical tools of statistical mechanics we find that the ecosystem diversity defined as the fraction of species that coexist at equilibrium decreases as the complexity i e number of characters of the species increases regardless of the modulating function by considering both selective and random elimination of the links in the community web we show that ecosystems composed of simple species are more robust than those composed of complex species in addition we show that the puzzling result that there exists either rich or poor ecosystems for a linear modulating function is not typical of communities in which the interspecies interactions are determined by a complementarity rule

on the meromorphic solutions of an equation of hayman abstract the behavior of meromorphic solutions of differential equations has been the subject of much study research has concentrated on the value distribution of meromorphic solutions and their rates of growth the purpose of the present paper is to show that a thorough search will yield a list of all meromorphic solutions of a multi parameter ordinary differential equation introduced by hayman this equation does not appear to be integrable for generic choices of the parameters so we do not find all solutions only those that are meromorphic this is achieved by combining wiman valiron theory and local series analysis hayman conjectured that all entire solutions of this equation are of finite order all meromorphic solutions of this equation are shown to be either polynomials or entire functions of order one

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

rigorous bounds of stokes constants for some nonlinear ordinary differential equations at rank one irregular singularities a rigorous way of obtaining sharp bounds for stokes constants is introduced and illustrated on a concrete problem arising in applications

the fisher hartwig formula and entanglement entropy toeplitz matrices have applications to different problems of statistical mechanics recently it was used for calculation of entanglement entropy in spin chains in the paper we review these recent developments we use the fisher hartwig formula as well as the recent results concerning the asymptotics of the block toeplitz determinants to calculate entanglement entropy of large block of spins in the ground state of xy spin chain

quantum spin chain toeplitz determinants and the fisher hartwig conjecture we consider the one dimensional quantum spin chain which is called the xx model xx0 model or isotropic xy model in a transverse magnetic field we are mainly interested in the entropy of a block of lneighboring spins at zero temperature and of an infinite system we represent the entropy in terms of a toeplitz determinant and calculate the asymptotic analytically we derive the first two terms of the asymptotic decomposition interestingly these two terms of decomposition clearly show a length scale related to the field h

painleve transcendent evaluations of finite system density matrices for 1d impenetrable bosons the recent experimental realisation of a one dimensional bose gas of ultra cold alkali atoms has renewed attention on the theoretical properties of the impenetrable bose gas of primary concern is the ground state occupation of effective single particle states in the finite system and thus the tendency for bose einstein condensation this requires the computation of the density matrix for the impenetrable bose gas on a circle we evaluate the density matrix in terms of a particular painleve vi transcendent in form and furthermore show that the density matrix satisfies a recurrence relation in the number of particles for the impenetrable bose gas in a harmonic trap and with dirichlet or neumann boundary conditions we give a determinant form for the density matrix a form as an average over the eigenvalues of an ensemble of random matrices and in special cases an evaluation in terms of a transcendent related to painleve v and vi we discuss how our results can be used to compute the ground state occupations

counting free fermions on a line a fisher hartwig asymptotic expansion for the toeplitz determinant in the double scaling limit we derive an asymptotic expansion for a wiener hopf determinant arising in the problem of counting one dimensional free fermions on a line segment at zero temperature this expansion is an extension of the result in the theory of toeplitz and wiener hopf determinants known as the generalized fisher hartwig conjecture the coefficients of this expansion are conjectured to obey certain periodicity relations which renders the expansion explicitly periodic in the counting parameter we present two methods to calculate these coefficients and verify the periodicity relations order by order the matrix riemann hilbert problem and the painleve v equation we show that the expansion coefficients are polynomials in the counting parameter and list explicitly first several coefficients

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

on linear systems and τ functions associated with lame s equation and painleve s equation vi abstract painleve s transcendental differential equation p vi may be expressed as the consistency condition for a pair of linear differential equations with 2 2 matrix coefficients with rational entries by a construction due to tracy and widom this linear system is associated with certain kernels which give trace class operators on hilbert space this paper expresses such operators in terms of hankel operators of linear systems which are realised in terms of the laurent coefficients of the solutions of the differential equations let p t l 2 0 l 2 t be the orthogonal projection then the fredholm determinant t det i p t defines the function which is here expressed in terms of the solution of a matrix gelfand levitan equation for suitable values of the parameters solutions of the hypergeometric equation give a linear system with similar properties for meromorphic transfer functions that have poles on an arithmetic progression the corresponding hankel operator has a simple form with respect to an exponential basis in l 2 0 so det i p t can be expressed as a series of finite determinants this applies to elliptic functions of the second kind such as satisfy lame s equation with l 1

the dependence on the monodromy data of the isomonodromic tau function the isomonodromic tau function defined by jimbo miwa ueno vanishes on the malgrange s divisor of generalized monodromy data for which a vector bundle is nontrivial or which is the same a certain riemann hilbert problem has no solution in their original work jimbo miwa ueno provided an algebraic construction of its derivatives with respect to isomonodromic times however the dependence on the generalized monodromy data i e monodromy representation and stokes parameters was not derived we fill the gap by providing a simpler and more general description in which all the parameters of the problem monodromy changing and monodromy preserving are dealt with at the same level we thus provide variational formulae for the isomonodromic tau function with respect to the generalized monodromy data the construction applies more generally given any sufficiently well behaved family of riemann hilbert problems rhp where the jump matrices depend arbitrarily on deformation parameters we can construct a one form not necessarily closed on the deformation space malgrange s differential defined off malgrange s divisor we then introduce the notion of discrete schlesinger transformation it means that we allow the solution of the rhp to have poles or zeros at prescribed point s even if is not closed its difference evaluated along the original solution and the transformed one is shown to be the logarithmic differential on the deformation space of a function as a function of the position of the points of the schlesinger transformation it yields a natural generalization of the sato formula for the baker akhiezer vector even in the absence of a tau function and it realizes the solution of the rhp as such ba vector some exemplifications in the setting of the painleve ii equation and finite toplitz hankel determinants are provided

a new approach to the self dual yang mills equations inspired by sato s new theory for soliton equations we find a new approach to the self dual yang mills equations we first establish a correspondence of solutions between the self dual yang mills equations and a new system of equations with infinitely many unknown functions it then turns out that the latter equations can be easily solved by a simple explicit procedure this leads to an explicit description of a very broad class of solutions to the self dual yang mills equations and also to a construction of transformations acting on these solutions

random replicators with high order interactions we use tools of the equilibrium statistical mechanics of disordered systems to study analytically the statistical properties of an ecosystem composed of n species interacting via random mutual interactions as well as via deterministic self interactions of order p 2 we show that the main effect of increasing the order of the interactions among the species is to make the system less competitive in the sense that the fraction of extinct species is greatly reduced in addition we find that for p 2 there is a threshold value which gives a lower bound to the concentration of the surviving species preventing then the existence of rare species and consequently increasing the robustness of the ecosystem to external perturbations

mean behaviour and distribution properties of multiplicative functions mean value theorems with sharp quantitative remainder term estimates and theorems on the distribution of values are proved for a large class of multiplicative functions this class is characterized by the existence of an abscissa of absolute convergence 1 for the dirichlet series of the quotient f s z s where f s is the dirichlet series associated with and z s denotes the riemann zeta function in particular the mean behaviour of on certain sequences a namely that of mersenne numbers 2 n 1 and that of shifted primes p 1 with natural n and prime p respectively is studied and cluster points of the image sequence f a of a are determined the strength of the results is demonstrated by new results for some specific functions

rank abundance relations in evolutionary dynamics of random replicators we present a nonequilibrium statistical mechanics description of rank abundance relations rar in random community models of ecology specifically we study a multispecies replicator system with quenched random interaction matrices we here consider symmetric interactions as well as asymmetric and antisymmetric cases rars are obtained analytically via a generating functional analysis describing fixed point states of the system in terms of a small set of order parameters and in dependence on the symmetry or otherwise of interactions and on the productivity of the community our work is an extension of tokita phys rev lett 93 178102 2004 where the case of symmetric interactions was considered within an equilibrium setup the species abundance distribution in our model come out as truncated normal distributions or transformations thereof and in some case are similar to left skewed distributions observed in ecology we also discuss the interaction structure of the resulting food web of stable species at stationarity cases of heterogeneous cooperation pressures as well as effects of finite system size and of higher order interactions

ecological communities from random generalized lotka volterra dynamics with nonlinear feedback we investigate the outcome of generalized lotka volterra dynamics of ecological communities with random interaction coefficients and nonlinear feedback we show in simulations that the saturation of nonlinear feedback stabilizes the dynamics this is confirmed in an analytical generating functional approach to generalized lotka volterra equations with piecewise linear saturating response for such systems we are able to derive self consistent relations governing the stable fixed point phase and to carry out a linear stability analysis to predict the onset of unstable behavior we investigate in detail the combined effects of the mean variance and covariance of the random interaction coefficients and the saturation value of the nonlinear response we find that stability and diversity increases with the introduction of nonlinear feedback where decreasing the saturation value has a similar effect to decreasing the covariance we also find cooperation to no longer have a detrimental effect on stability with nonlinear feedback and the order parameters mean abundance and diversity to be less dependent on the symmetry of interactions with stronger saturation

random words toeplitz determinants and integrable systems ii this paper connects the analysis of the length of the longest weakly increasing subsequence of inhomogeneous random words to a riemnn hilbert problem and an associated system of integrable partial differential equations in particular we show that the poissonization of the distribution function of this length can be identified as the jimbo miwa ueno tau function

eigenpairs of toeplitz and disordered toeplitz matrices with a fisher hartwig symbol toeplitz matrices have entries that are constant along diagonals they model directed transport are at the heart of correlation function calculations of the two dimensional ising model and have applications in quantum information science we derive their eigenvalues and eigenvectors when the symbol is singular fisher hartwig we then add diagonal disorder and study the resulting eigenpairs we find that there is a bulk behavior that is well captured by second order perturbation theory of non hermitian matrices the non perturbative behavior is classified into two classes runaways type i leave the complex valued spectrum and become completely real because of eigenvalue attraction runaways type ii leave the bulk and move very rapidly in response to perturbations these have high condition numbers and can be predicted localization of the eigenvectors are then quantified using entropies and inverse participation ratios eigenvectors corresponding to runaways type ii are most localized i e super exponential whereas runaways type i are less localized than the unperturbed counterparts and have most of their probability mass in the interior with algebraic decays the results are corroborated by applying free probability theory and various other supporting numerical studies

an area preserving action of the modular group on cubic surfaces and the painleve vi equation we construct an area preserving action of the modular group on a general 4 parameter family of affine cubic surfaces we present a geometrical background behind this construction that is a natural symplectic structure on a moduli space of rank two linear monodromy representations over the 2 dimensional sphere with four punctures and a natural symplectic action upon it of the braid group on three strings studying this action as a discrete dynamical system will be important in discussing the monodromy of the painleve vi equation

toeplitz determinants with one fisher hartwig singularity letcbe a function defined on the unit circle with fourier coefficients cn n the fisher hartwig conjecture describes the asymptotic behaviour of the determinants of then ntoeplitz matricesdn c det ci j n 1i j 0for a certain class of functionsc in this paper we prove this conjecture in the case of functions with one singularity more precisely we consider functions of the formc ei b ei t ei 1 u ei 1 heret ei exp i 0 2 is a function with a jump discontinuity u ei 2 2 cos is a function which may have a zero a pole or a discontinuity of oscillating type andbis a sufficiently smooth nonvanishing function with winding number equal to zero the only restriction we impose on the parameters is that 2 is required not to be a negative integer in the case where re 1 2 i e where the corresponding functioncis not integrable we identifycin an appropriate way with a distribution

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

evolutionary dynamics of populations with conflicting interactions classification and analytical treatment considering asymmetry and power evolutionary game theory has been successfully used to investigate the dynamics of systems in which many entities have competitive interactions from a physics point of view it is interesting to study conditions under which a coordination or cooperation of interacting entities will occur be it spins particles bacteria animals or humans here we analyze the case where the entities are heterogeneous particularly the case of two populations with conflicting interactions and two possible states for such systems explicit mathematical formulas will be determined for the stationary solutions and the associated eigenvalues which determine their stability in this way four different types of system dynamics can be classified and the various kinds of phase transitions between them will be discussed while these results are interesting from a physics point of view they are also relevant for social economic and biological systems as they allow one to understand conditions for 1 the breakdown of cooperation 2 the coexistence of different behaviors subcultures 3 the evolution of commonly shared behaviors norms and 4 the occurrence of polarization or conflict we point out that norms have a similar function in social systems that forces have in physics

random replicators with asymmetric couplings systems of interacting random replicators are studied using generating functional techniques while replica analyses of such models are limited to systems with symmetric couplings dynamical approaches as presented here allow us specifically to address cases with asymmetric interactions where there is no lyapunov function governing the dynamics we focus on replicator models with gaussian couplings of general symmetry between p 2 species and discuss how an effective description of the dynamics can be derived in terms of a single species process upon making a fixed point ansatz persistent order parameters in the ergodic stationary states can be extracted from this process and different types of phase transitions can be identified and related to each other we discuss the effects of asymmetry in the couplings on the order parameters and the phase behaviour for p 2 and p 3 numerical simulations verify our theory for the case of cubic interactions numerical experiments indicate regimes in which only a finite number of species survives even when the thermodynamic limit is considered

spectral functions special functions and the selberg zeta function the functional determinant of an eigenvalue sequence as defined by zeta regularization can be simply evaluated by quadratures we apply this procedure to the selberg trace formula for a compact riemann surface to find a factorization of the selberg zeta function into two functional determinants respectively related to the laplacian on the compact surface itself and on the sphere we also apply our formalism to various explicit eigenvalue sequences reproducing in a simpler way classical results about the gamma function and the barnesg function concerning the latter our method explains its connection to the selberg zeta function and evaluates the related glaisher kinkelin constanta

the cauchy problem for the second member of a rm p_ iv hierarchy a rigorous method the inverse monodromy transform for studying the riemann hilbert rh problem associated with the classical painleve equations is applied to the second member of a fourth painleve hierarchy we show that the cauchy problem for the second member of this hierarchy admits in general a global meromorphic solution in x moreover for a particular choice of the monodromy data the associated rh problem can be reduced to a set of scalar rh problems and a special solution which can be written in terms of the airy function is obtained

a new correlator in quantum spin chains we propose a new correlator in one dimensional quantum spin chains the s emptiness formation probability s efp this is a generalization of the emptiness formation probability efp which is the probability that the first n spins of the chain are all aligned downwards in the s efp we let the spins in question be separated by s sites the usual efp corresponds to the special case when s 1 taking s 1 allows us to quantify non local correlations we express the s efp for the anisotropic xy model in a transverse magnetic field a system with both critical and non critical regimes in terms of a toeplitz determinant for the isotropic xy model we find that the magnetic field induces an interesting length scale

transformations rs42 3 of the ranks 4 and algebraic solutions of the sixth painleve equation compositions of rational transformations of independent variables of linear matrix ordinary differential equations odes with the schlesinger transformations rs transformations are used to construct algebraic solutions of the sixth painleve equation rs transformations of the ranks 3 and 4 of 2 2 matrix fuchsian odes with 3 singular points into analogous ode with 4 singular points are classified

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

toeplitz matrices and determinants with fisher hartwig symbols abstract this paper is concerned with toeplitz matrices generated by symbols of the form a t r 1 r t t r 2 r t t r r b t t 1 where t 1 t r are pairwise distinct points on the unit circle b is sufficiently smooth b t 0 t 1 and ind b 0 t t r r is defined as exp i r arg t t r with arg t t r and r r are complex numbers satisfying re r 1 2 re r 1 2 weighted hardy spaces l 2 1 and l 2 2 are defined such that the toeplitz operator t a l 2 1 l 2 2 is bounded and invertible and that the finite section method is applicable to t a considered as acting from l 2 1 onto l 2 2 these results are then applied to prove a conjecture of fisher and hartwig adv in chem phys 15 1968 333 353 which asserts that the determinants of the n n sections of t a are asymptotically equal to g b n n q e a n where q r 2 r 2 and g b e a are certain nonzero constants under quite general conditions on the smoothness of b these constants are completely identified

log gases random matrices and the fisher hartwig conjecture some features of the probability e n r of a region r in certain log potential systems containing precisely n particles are noted first it is shown that a quantity analogous to e n r for a new solvable two component log gas can be expressed in terms of the toeplitz determinant discretization of a fredholm determinant which occurs in the calculation of e n r for hermitian random matrices second the first two terms of the asymptotic large r expansion of e n r for complex random matrices when r is a disk are derived by using an electrostatic thermodynamic argument based on an analogy with the two dimensional one component plasma finally by using the fisher hartwig conjecture from the theory of toeplitz determinants the asymptotics of e 0 r for a class of one dimensional lattice systems is shown to obey a sum rule which has been conjectured to be applicable to all fluid systems with exclusively mobile species

the indefinite anti self dual metrics and the painleve equations we classify the su 2 invariant anti self dual metrics with a signature the metrics are specified by a solution of painleve vi v iii or ii moreover we show the geometric meaning of the metrics specified by each type of painleve function

parameterized picard vessiot extensions and atiyah extensions abstract generalizing atiyah extensions we introduce and study differential abelian tensor categories over differential rings by a differential ring we mean a commutative ring with an action of a lie ring by derivations in particular these derivations act on a differential category a differential tannakian theory is developed the main application is to the galois theory of linear differential equations with parameters namely we show the existence of a parameterized picard vessiot extension and therefore the galois correspondence for many differential fields with possibly non differentially closed fields of constants that is fields of functions of parameters other applications include a substantially simplified test for a system of linear differential equations with parameters to be isomonodromic which will appear in a separate paper this application is based on differential categories developed in the present paper and not just differential algebraic groups and their representations

quantum painleve calogero correspondence the painleve calogero correspondence is extended to auxiliary linear problems associated with painleve equations the linear problems are represented in a new form which has a suggestive interpretation as a quantized version of the painleve calogero correspondence namely the linear problem responsible for the time evolution is brought into the form of non stationary schrodinger equation in imaginary time t 12 x2 v x t whose hamiltonian is a natural quantization of the classical calogero like hamiltonian h 12p2 v x t for the corresponding painleve equation in present paper we present explicit constructions for the first five equations from the painleve list

on the central quadric ansatz integrable models and painleve reductions it was observed by tod 1995 class quantum grav 12 1535 47 and later by dunajski and tod 2002 phys lett a 303 253 64 that the boyer finley bf and the dispersionless kadomtsev petviashvili dkp equations possess solutions whose level surfaces are central quadrics in the space of independent variables the so called central quadric ansatz it was demonstrated that generic solutions of this type are described by painlev equations piii and pii respectively the aim of our paper is threefold 1 based on the method of hydrodynamic reductions we classify integrable models possessing the central quadric ansatz this leads to the five canonical forms including bf and dkp 2 applying the central quadric ansatz to each of the five canonical forms we obtain all painlev equations pi pvi with pvi corresponding to the generic case of our classification 3 we argue that solutions coming from the central quadric ansatz constitute a subclass of two phase solutions provided by the method of hydrodynamic reductions

the fisher hartwig conjecture and generalizations abstract we discuss the status of the fisher hartwig conjecture concerning the asymptotic expansion of a class of toeplitz determinants with singular generating functions a counterexample is given for a nonrational generating function and we formulate a generalized fisher hartwig conjecture

second order catalytic quasispecies yields discontinuous mean fitness at error threshold the quasispecies model describes processes related to the origin of life and viral evolutionary dynamics we discuss how the error catastrophe that reflects the transition from localized to delocalized quasispecies population is affected by catalytic replication of different reaction orders specifically we find that second order mechanisms lead to a discontinuity in the mean fitness of the population at the error threshold this is in contrast to the behavior of the first order autocatalytic replication mechanism considered in the standard quasispecies model this suggests that quasispecies models with higher order replication mechanisms produce discontinuities in the mean fitness and hence the viable population fraction as well at the error threshold while lower order replication mechanisms yield a continuous mean fitness function we discuss potential implications for understanding replication in the rna world and in virology

the k tacnode process the tacnode process is a universal behavior arising in nonintersecting particle systems and tiling problems for dyson brownian bridges the tacnode process describes the grazing collision of two packets of walkers we consider such a dyson sea on the unit circle with drift for any k in mathbb z we show that an appropriate double scaling of the drift and return time leads to a generalization of the tacnode process in which k particles are expected to wrap around the circle we derive winding number probabilities and an expression for the correlation kernel in terms of functions related to the generalized hastings mcleod solutions to the inhomogeneous painleve ii equation the method of proof is asymptotic analysis of discrete orthogonal polynomials with a complex weight

asymptotics of toeplitz determinants and the emptiness formation probability for the xy spin chain we study an asymptotic behaviour of a special correlator known as the emptiness formation probability efp for the one dimensional anisotropic xy spin 1 2 chain in a transverse magnetic field this correlator is essentially the probability of formation of a ferromagnetic string of length n in the antiferromagnetic ground state of the chain and plays an important role in the theory of integrable models for the xy spin chain the correlator can be expressed as the determinant of a toeplitz matrix and its asymptotical behaviours for n throughout the phase diagram are obtained using known theorems and conjectures on toeplitz determinants we find that the decay is exponential everywhere in the phase diagram of the xy model except on the critical lines i e where the spectrum is gapless in these cases a power law prefactor with a universal exponent arises in addition to an exponential or gaussian decay the latter gaussian behaviour holds on the critical line corresponding to the isotropic xy model while at the critical value of the magnetic field the efp decays exponentially at small anisotropy one has a crossover from the gaussian to the exponential behaviour we study this crossover using the bosonization approach