classification of integrable hydrodynamic chains and generating functions of conservation laws a new approach to classification of integrable hydrodynamic chains is established generating functions of conservation laws are classified by the method of hydrodynamic reductions the n parametric family of explicit hydrodynamic reductions allows us to reconstruct corresponding hydrodynamic chains many new hydrodynamic chains are found

on a class of three dimensional integrable lagrangians we characterize non degenerate lagrangians of the form such that the corresponding euler lagrange equations are integrable by the method of hydrodynamic reductions the integrability conditions constitute an over determined system of fourth order pdes for the lagrangian density f which is in involution and possesses interesting differential geometric properties the moduli space of integrable lagrangians factorized by the action of a natural equivalence group is three dimensional familiar examples include the dispersionless kadomtsev petviashvili dkp and the boyer finley lagrangians f u x 3 3 u y 2 u x u t and respectively a complete description of integrable cubic and quartic lagrangians is obtained up to the equivalence transformations the list of integrable cubic lagrangians reduces to three examples there exists a unique integrable quartic lagrangian we conjecture that these examples exhaust the list of integrable polynomial lagrangians which are essentially three dimensional it was verified that there exist no polynomial integrable lagrangians of degree five we prove that the euler lagrange equations are integrable by the method of hydrodynamic reductions if and only if they possess a scalar pseudopotential playing the role of a dispersionless lax pair

hydrodynamic chains and the classification of their poisson brackets infinite component poisson brackets of the dubrovin novikov type sov math dokl 27 665 669 1983 are considered the corresponding jacobi identity is significantly simplified in the liouville coordinates since the skew symmetry condition is automatically satisfied the concept of m poisson bracket connected with hydrodynamic chains is introduced then the jacobi identity is a nonlinear system of equations in partial derivatives which can be completely integrated in such a case a classification of infinite component poisson brackets of the dubrovin novikov type can be obtained two simplest examples m 0 and m 1 are considered also infinite component poisson brackets of the ferapontov am math soc transl 170 38 58 1995 type can be simplified in the liouville coordinates the jacobi identity for infinite component poisson brackets of the ferapontov mokhov type russ math surveys 45 218 219 1990 is presented in the liouville coordinates

classical r matrix theory of dispersionless systems ii 2 1 dimension theory a systematic way of constructing 2 1 dimensional dispersionless integrable hamiltonian systems is presented the method is based on the so called central extension procedure and classical r matrix applied to the poisson algebras of formal laurent series results are illustrated with the known and new 2 1 dimensional dispersionless systems

r matrix lax pair and multiparameter decompositions of lie algebras we construct r operators on the lie algebra x1d524 x1d529 n or more generally hilbert schmidt operators l2 in hilbert space these operators are related to a multiparameter deformation given by a sequence of parameters a1 a2 we determine for which choices of parameters r operators are r matrices we also construct the lax pair for the corresponding hamilton equations

solving the dispersionless dym equation by hodograph transformations the method of hodograph transformation is used to find a class of exact solutions of the dispersionless dym ddym equation and its reductions in particular we discuss hodograph solutions of one variable and two variable reductions of the ddym system and investigate their associated lax as well as hamiltonian formalism

direct evaluation of large deviation functions we introduce a numerical procedure to evaluate directly the probabilities of large deviations of physical quantities such as current or density that are local in time the large deviation functions are given in terms of the typical properties of a modified dynamics and since they no longer involve rare events can be evaluated efficiently and over a wider ranges of values we illustrate the method with the current fluctuations of the totally asymmetric exclusion process and with the work distribution of a driven lorentz gas

classical r matrix theory for bi hamiltonian field systems this is a survey of the application of the classical r matrix formalism to the construction of infinite dimensional integrable hamiltonian field systems the main point is the study of bi hamiltonian structures appropriate constructions on poisson noncommutative and loop algebras as well as the central extension procedure are presented the theory is developed for 1 1 and 2 1 dimensional field and lattice soliton systems as well as hydrodynamic systems the formalism presented contains sufficiently many proofs and important details to make it self contained and complete the general theory is applied to several infinite dimensional lie algebras in order to construct both dispersionless and dispersive soliton integrable field systems

on the integrability of 2 1 dimensional quasilinear systems a 2 1 dimensional quasilinear system is said to be integrable if it can be decoupled in infinitely many ways into a pair of compatible n component one dimensional systems in riemann invariants exact solutions described by these reductions known as nonlinear interactions of planar simple waves can be viewed as natural dispersionless analogues of n gap solutions it is demonstrated that the requirement of the existence of sufficiently many n component reductions provides the effective classification criterion as an example of this approach we classify integrable 2 1 dimensional systems of conservation laws possessing a convex quadratic entropy

finite time current probabilities in the asymmetric exclusion process on a ring we calculate the time dependent probability distribution of current through a selected bond in the totally asymmetric exclusion process with periodic boundary conditions we derive a general formula for the probability that the integrated current exceeds a given value n at the moment of time t the formula is written in a form of a contour integral of a determinant of a toeplitz matrix transforming the determinant expression we obtain a generalization of the known formula derived by johansson for the infinite one dimensional lattice to check the general formula we consider the specific case corresponding to the probability of a minimal non zero current for this case we get an explicit analytical expression and analyze its asymptotics

the complex zeros of random polynomials mark kac gave an explicit formula for the expectation of the number vn a of zeros of a random polynomial n i pn z e tj j o in any measurablc subset q of the reals here in i are independent standard normal random variables in fact for each n 1 he obtained an explicit intensity function gn for which e vn l jgn x dx here we extend this formula to obtain an explicit formula for the expected number of zeros in any measurable subset q of the complex plane c namely we show that e vn ki j hn x y dxdy j gn x dx where hn is an explicit intensity function we also study the asymptotics of hn showing that for large n its mass lies close to and is uniformly distributed around the unit circle

on deformations of standard r matrices for integrable infinite dimensional systems simple deformations with a parameter of classical r matrices which follow from decomposition of appropriate lie algebras are considered as a result nonstandard lax representations for some well known integrable systems are presented and integrable evolution equations are constructed

lie algebraic approach to the construction of 2 1 dimensional lattice field and field integrable hamiltonian equations two different methods for the construction of 2 1 dimensional integrable lattice field and field hamiltonian dynamical systems are presented the first method is based on the so called central extension procedure applied to the lie algebra of shift operators and the lie algebra of pseudodifferential operators the second method is the so called operand formalism both methods allow a construction of some new integrable nonlinear hamiltonian lattice field and field equations in 2 1 dimensional space

classical r matrices and compatible poisson structures for lax equations on poisson algebras given a classical r matrix on a poisson algebra we show how to construct a natural family of compatible poisson structures for the hamiltonian formulation of lax equations examples for which our formalism applies include the benny hierarchy the dispersionless toda lattice hierarchy the dispersionless kp and modified kp hierarchies the dispersionless dym hierarchy etc

density of complex critical points of a real random so m 1 polynomial we study the density of complex critical points of a real random so m 1 polynomial in m variables in a previous paper macdonald in j stat phys 136 5 807 2009 the author used the poincare lelong formula to show that the density of complex zeros of a system of these real random polynomials rapidly approaches the density of complex zeros of a system of the corresponding complex random polynomials the su m 1 polynomials in this paper we use the kac rice formula to prove an analogous result the density of complex critical points of one of these real random polynomials rapidly approaches the density of complex critical points of the corresponding complex random polynomial in one variable we give an exact formula and a scaling limit formula for the density of critical points of the real random so 2 polynomial as well as for the density of critical points of the corresponding complex random su 2 polynomial

distribution of zeros of random and quantum chaotic sections of positive line bundles we study the limit distribution of zeros of certain sequences of holomorphic sections of high powers m n of a positive holomorphic hermitian line bundle l over a compact complex manifold m our first result concerns random sequences of sections using the natural probability measure on the space of sequences of orthonormal bases s n j of h 0 m l n we show that for almost every sequence s n j the associated sequence of zero currents 1 n z s n j tends to the curvature form of l thus the zeros of a sequence of sections s n h 0 m l n chosen independently and at random become uniformly distributed our second result concerns the zeros of quantum ergodic eigenfunctions where the relevant orthonormal bases s n j of h 0 m l n consist of eigensections of a quantum ergodic map we show that also in this case the zeros become uniformly distributed

an extended harry dym hierarchy an extended harry dym hierarchy is constructed by using eigenfunctions and adjoint eigenfunctions of the spectral problems of the harry dym hierarchy associated with the pseudo differential operator l u u0 u1 1 the corresponding lax presentation possesses a self consistent source involving squared eigenfunctions the resulting extended harry dym hierarchy is reduced to the harry dym hierarchy with self consistent sources under the n reduction ln ln 2 and the k constrained harry dym hierarchy under the k constraint lk lk 2 ni 1qi 1ri 2 a few particular examples are computed together with their lax pairs

on dispersionless hirota equations of the dispersionless dym hierarchy we show that under twistor construction the only finite dimensional reduction of the dispersionless dym hierarchy which can be characterized by a free energy is the two reduction model with lax operator of the form up u0 p 1 the primary variables u and u0 can be expressed in terms of second derivatives of the associated free energy and the hierarchy flows can be written as a set of dispersionless hirota equations

chaotic analytic zero points exact statistics for those of a random spin state a natural statistical ensemble of 2j points on the unit sphere can be associated via the majorana representation with a random quantum state of spin j and an exact expression is obtained here for the general k point correlation function in this ensemble the pair correlation in the large j limit takes the relatively simple form where and is the angular separation of the pair of points on the sphere it appears from the numerical work of others that in this limit these statistics are typical of the zero points of analytic functions associated with chaotic quantum dynamical systems

distribution of roots of random polynomials we consider polynomials of high degree with random coefficients which appear in the context of quantum chaotic dynamics and investigate various conditions under which their roots tend to concentrate near the unit circle in the complex plane correlation functions of roots are computed analytically we also investigate a certain class of random polynomials whose roots cover in a uniform way the riemann sphere special emphasis is devoted to the influence of symmmetries

bethe ansatz for the weakly asymmetric simple exclusion process and phase transition in the current distribution the probability distribution of the current in the asymmetric simple exclusion process is expected to undergo a phase transition in the regime of weak asymmetry of the jumping rates this transition was first predicted by bodineau and derrida using a linear stability analysis of the hydrodynamical limit of the process and further arguments have been given by mallick and prolhac however it has been impossible so far to study what happens after the transition the present paper presents an analysis of the large deviation function of the current on both sides of the transition from a bethe ansatz approach of the weak asymmetry regime of the exclusion process

from dispersionless to soliton systems via weyl moyal like deformations the formalism of quantization deformation is reviewed and the weyl moyal like deformation is applied to systematic construction of the field and lattice integrable soliton systems from poisson algebras of dispersionless systems

poincare lelong approach to universality and scaling of correlations between zeros this note is concerned with the scaling limit as n of n point correlations between zeros of random holomorphic polynomials of degree n in m variables more generally we study correlations between zeros of holomorphic sections of powers ln of any positive holomorphic line bundle l over a compact kahler manifold distances are rescaled so that the average density of zeros is independent of n our main result is that the scaling limits of the correlation functions and more generally of the correlation forms are universal i e independent of the bundle l manifold m or point on m

s functions reductions and hodograph solutions of the rth dispersionless modified kp and dym hierarchies we introduce an s function formulation for the recently found rth dispersionless modified kp and rth dispersionless dym hierarchies giving also a connection of these s functions with the orlov functions of the hierarchies then we discuss a reduction scheme for the hierarchies that together with the s function formulation leads to hodograph systems for the associated solutions we consider also the connection of these reductions with those of the dispersionless kp hierarchy and with hydrodynamic type systems in particular for the one component and two component reduction we derive for both hierarchies ample sets of examples of explicit solutions

large deviation function of the partially asymmetric exclusion process the large deviation function obtained recently by derrida and lebowitz phys rev lett 80 209 1998 for the totally asymmetric exclusion process is generalized to the partially asymmetric case in the scaling limit the asymmetry parameter rescales the scaling variable in a simple way the finite size corrections to the universal scaling function and the universal cumulant ratio are also obtained to the leading order

dressing operator approach to the toroidal model of higher dimensional dispersionless kp hierarchy based upon the dressing operator approach we investigate the twistor theoretical construction additional symmetries hodograph solutions and miura transformation for the toroidal model of higher dimensional dispersionless kp hierarchy introduced by takasaki

exact statistics of complex zeros for gaussian random polynomials with real coefficients k point correlations of complex zeros for gaussian ensembles of random polynomials of order n with real coefficients grprc are calculated exactly following an approach of hannay for the case of gaussian random polynomials with complex coefficients grpcc it is shown that in the thermodynamic limit of gaussian random holomorphic functions all the statistics converge to their grpcc counterparts as one moves off the real axis while close to the real axis the two cases are essentially different special emphasis is given to one and two point correlation functions in various regimes

algebraic structure of the gradient holonomic algorithm for lax integrable nonlinear dynamical systems i the algebraic structure of the gradient holonomic algorithm for lax integrable dynamical systems is discussed a generalization of the r structure approach for the case of operator valued affine lie algebras is used to prove the bi hamiltonian formulation of nonlinear integrable dynamical systems in multidimensions the monodromy transfer matrix is constructed to describe the operator manifold in relation to canonical lie poisson bracket on initial affine lie algebra with gauge central extension as an illustration the two dimensional operator benney kaup integrable hierarchy is considered and their bi hamiltonicity is proved

dispersionless 3 1 dimensional integrable hierarchies in this paper we introduce a multi dimensional version of the r matrix approach to the construction of integrable hierarchies applying this method to the case of the lie algebra of functions with respect to the contact bracket we construct integrable hierarchies of 3 1 dimensional dispersionless systems of the type recently introduced in sergyeyev 2014 http arxiv org abs 1401 2122

universal large deviation function of the kardar parisi zhang equation in one dimension using the bethe ansatz we calculate the whole large deviation function of the displacement of particles in the asymmetric simple exclusion process asep on a ring when the size of the ring is large the central part of this large deviation function takes a scaling form independent of the density of particles we suggest that this scaling function found for the asep is universal and should be characteristic of all the systems described by the kardar parisi zhang equation in 1 1 dimension simulations done on two simple growth models are in reasonable agreement with this conjecture

critical points and supersymmetric vacua i supersymmetric vacua universes of string m theory may be identified with certain critical points of a holomorphic section the superpotential of a hermitian holomorphic line bundle over a complex manifold an important physical problem is to determine how many vacua there are and how they are distributed as the superpotential varies over physically relevant ensembles in several papers over the last few years m r douglas and co workers have studied such vacuum statistics problems for a variety of physical models at the physics level of rigor do ad dd the present paper is the first of a series by the present authors giving a rigorous mathematical foundation for the vacuum statistics problem it sets down basic results on the statistics of critical points s 0 of random holomorphic sections of hermitian holomorphic line bundles with respect to a metric connection when the sections are endowed with a gaussian measure the principal results give formulas for the expected density and number of critical points of fixed morse index of gaussian random sections relative to they are particularly concrete for riemann surfaces in our subsequent work the results will be applied to the vacuum statistics problem and to the purely geometric problem of studying metrics which minimize the expected number of critical points

dispersive deformations of hamiltonian systems of hydrodynamic type in 2 1 dimensions abstract we develop a theory of integrable dispersive deformations of 2 1 dimensional hamiltonian systems of hydrodynamic type following the scheme proposed by dubrovin and his collaborators in 1 1 dimensions our results show that the multi dimensional situation is far more rigid and generic hamiltonians are not deformable as an illustration we discuss a particular class of two component hamiltonian systems establishing the triviality of first order deformations and classifying hamiltonians possessing nontrivial deformations of the second order

on roots of random polynomials we study the distribution of the complex roots of random polynomials of degree n with i i d coefficients using techniques related to rice s treatment of the real roots question we derive under appropriate moment and regularity conditions an exact formula for the average density of this distribution which yields appropriate limit average densities further using a different technique we prove limit distribution results for coefficients in the domain of attraction of the stable law

a note on the dispersionless bkp hierarchy we discuss the miura map as well as the poisson algebras associated with the dispersionless dym hierarchy particularly we study explicitly the bi hamiltonian structure of a truncated dym system with two variables in which a new hierarchy flow generated by logarithmic hamiltonians appears we then show that this new hierarchy emerges naturally from the topological recursion relation in the landau ginzburg formulation

quantum chaotic dynamics and random polynomials we investigate the distribution of roots of polynomials of high degree with random coefficients which among others appear naturally in the context of quantum chaotic dynamics it is shown that under quite general conditions their roots tend to concentrate near the unit circle in the complex plane in order to further increase this tendency we study in detail the particular case of self inversive random polynomials and show that for them a finite portion of all roots lies exactly on the unit circle correlation functions of these roots are also computed analytically and compared to the correlations of eigenvalues of random matrices the problem of ergodicity of chaotic wavefunctions is also considered for that purpose we introduce a family of random polynomials whose roots spread uniformly over phase space while these results are consistent with random matrix theory predictions they provide a new and different insight into the problem of quantum ergodicity special attention is devoted to the role of symmetries in the distribution of roots of random polynomials

universality and scaling of correlations between zeros on complex manifolds we study the limit as n of the correlations between simultaneous zeros of random sections of the powers l n of a positive holomorphic line bundle l over a compact complex manifold m when distances are rescaled so that the average density of zeros is independent of n we show that the limit correlation is independent of the line bundle and depends only on the dimension of m and the codimension of the zero sets we also provide some explicit formulas for pair correlations in particular we prove that hannay s limit pair correlation function for su 2 polynomials holds for all compact riemann surfaces

pfaffian point process for the gaussian real generalised eigenvalue problem the generalised eigenvalues for a pair of n n matrices x1 x2 are defined as the solutions of the equation det x1 x2 0 or equivalently for x2 invertible as the eigenvalues of x 2 1 x 1 we consider gaussian real matrices x1 x2 for which the generalised eigenvalues have the rotational invariance of the half sphere or after a fractional linear transformation the rotational invariance of the unit disk in these latter variables we calculate the joint eigenvalue probability density function the probability pn k of finding k real eigenvalues the densities of real and complex eigenvalues the latter being related to an average over characteristic polynomials and give an explicit pfaffian formula for the higher correlation functions rho k 1 k 2 a limit theorem for pn k is proved and the scaled form of rho k 1 k 2 is shown to be identical to the analogous limit for the correlations of the eigenvalues of real gaussian matrices we show that these correlations satisfy sum rules characteristic of the underlying two component coulomb gas

the haantjes tensor and double waves for multi dimensional systems of hydrodynamic type a necessary condition for integrability an invariant differential geometric approach to the integrability of 2 1 dimensional systems of hydrodynamic type is developed we prove that the existence of special solutions known as double waves is equivalent to the diagonalizability of an arbitrary matrix of the two parameter familysince the diagonalizability can be effectively verified by calculating the haantjes tensor this provides a simple necessary condition for integrability

algebraic structure of the gradient holonomic algorithm for lax integrable nonlinear dynamical systems ii the reduction via dirac and canonical quantization procedure the generalized theory of the r structure on affine operator lie algebras is used to construct a complete theory of lax integrable nonlinear dynamical systems in multidimensions the operator bi hamiltonian structures and their functional reductions are discussed in great detail in the examples of operator korteweg de vries and benney kaup dynamical systems as an important by product of the developed algebraic theory the dirac canonical quantization problem is solved almost completely for the neumann bogoliubov type oscillatory dynamical system on spheres associated via moser with the spectral moment map on an affine associative metrized lie coalgebra with a one parameter gauge two cocycle some remarks are given on the problem of extending the developed algebraic theory to the case of lax integrable dynamical systems on discrete manifolds

dispersionful analog of the whitham hierarchy the transition from the zero genus universal whitham hierarchy to its dispersionful counterpart making use only of the lax representations is presented this is an alternative approach to that of takasaki who has recently shown that the dispersionless limit of the charged multicomponent kadomtsev petviashvili kp hierarchy is the whitham hierarchy the advantage of the presented approach is the construction of finite field reductions which are the main concern in this paper the theory is illustrated by several significant examples

meromorphic lax representations of 1 1 dimensional multi hamiltonian dispersionless systems rational lax hierarchies introduced by krichever are generalized a systematic construction of infinite multi hamiltonian hierarchies and related conserved quantities is presented the method is based on the classical r matrix approach applied to poisson algebras a proof that poisson operators constructed near different points of laurent expansion of lax functions are equal is given all results are illustrated by several examples

how many zeros of a random polynomial are real we provide an elementary geometric derivation of the kac integral formula for the expected number of real zeros of a random polynomial with independent standard normally distributed coefficients we show that the expected number of real zeros is simply the length of the moment curve 1 t tn projected onto the surface of the unit sphere divided by the probability density of the real zeros is proportional to how fast this curve is traced out we then relax kac s assumptions by considering a variety of random sums series and distributions and we also illustrate such ideas as integral geometry and the fubini study metric

stochastic burgers and kpz equations from particle systems we consider two strictly related models a solid on solid interface growth model and the weakly asymmetric exclusion process both on the one dimensional lattice it has been proven that in the diffusive scaling limit the density field of the weakly asymmetric exclusion process evolves according to the burgers equation and the fluctuation field converges to a generalized ornstein uhlenbeck process we analyze instead the density fluctuations beyond the hydrodynamical scale and prove that their limiting distribution solves the non linear burgers equation with a random noise on the density current for the solid on solid model we prove that the fluctuation field of the interface profile if suitably rescaled converges to the kardar parisi zhang equation this provides a microscopic justification of the so called kinetic roughening i e the non gaussian fluctuations in some non equilibrium processes our main tool is the cole hopf transformation and its microscopic version we also develop a mathematical theory for the macroscopic equations

density of complex zeros of a system of real random polynomials we study the density of complex zeros of a system of real random so m 1 polynomials in m variables we show that the density of complex zeros of this random polynomial system with real coefficients rapidly approaches the density of complex zeros in the complex coefficients case we also show that the behavior the scaled density of complex zeros near m of the system of real random polynomials is different in the m 2 case than in the m 1 case the density approaches infinity instead of tending linearly to zero

poisson algebras associated with constrained dispersionless modified kadomtsev petviashvili hierarchies we investigate the bi hamiltonian structures associated with constrained dispersionless modified kadomtsev petviashvili kp hierarchies which are constructed from truncations of the lax operator of the dmkp hierarchy after transforming their second hamiltonian structures to those of the gelfand dickey type we obtain the poisson algebras of the coefficient functions of the truncated lax operators then we study the conformal property and free field realizations of these poisson algebras some examples are worked out explicitly to illustrate the obtained results

integrable discrete systems on r and related dispersionless systems a general framework for integrable discrete systems on r in particular containing lattice soliton systems and their q deformed analogs is presented the concept of regular grain structures on r generated by discrete one parameter groups of diffeomorphisms in terms of which one can define algebra of shift operators is introduced two integrable hierarchies of discrete chains together with bi hamiltonian structures and their continuous limits are constructed the inverse problem based on the deformation quantization scheme is considered

integrable hydrodynamic chains a new approach for derivation of benney type moment chains and integrable hydrodynamic type systems is presented new integrable hydrodynamic chains are constructed all their hydrodynamical reductions are described and integrated new 2 1 integrable hydrodynamic type systems are found