integration of quantum many body problems by affine knizhnik zamolodchikov equations abstract an equivalence of the affine analogues of knizhnik zamolodchikov and dunkl equations is established for arbitrary root systems as a corollary an isomorphism is obtained from the solutions of the above kz equations in the induced representations of hecke algebras to the eigenfunctions for the corresponding matrix quantum calogero sutherland operators which generalizes the matsuo map

the multicomponent kp hierarchy differential fay identities and lax equations in this paper we show that four sets of differential fay identities of an n component kp hierarchy derived from the bilinear relation satisfied by the tau function of the hierarchy are sufficient to derive the auxiliary linear equations for the wavefunctions from this we derive the lax representation for the n component kp hierarchy which are equations satisfied by some pseudo differential operators with matrix coefficients besides the lax equations with respect to the time variables proposed in date et al 1981 j phys soc japan 50 3806 12 we also obtain a set of equations relating different charge sectors which can be considered as a generalization of the modified kp hierarchy proposed in takebe 2002 lett math phys 59 157 72

optimal nonzero set point regulation via fixed order dynamic compensation standard lqg linear quadratic gaussian theory is generalized to a regulation problem involving specified nonzero setpoints for the state and control variables and nonzero mean disturbances for generality the results are obtained for the problem of fixed order i e not necessarily full order dynamic compensation when the state control and disturbance offsets are set to zero and the compensator order is set equal to the plant dimension the standard lqg result is recovered these results provide the dynamic counterpart for the nonzero set point regulation results obtained by d s bernstein and w m haddad 1987 by means of static controllers

backlund transformations and hamiltonian flows in this work we show that under certain conditions parametric backlund transformations for a finite dimensional integrable system can be interpreted as solutions to the equations of motion defined by an associated non autonomous hamiltonian the two systems share the same constants of motion this observation leads to the identification of the hamiltonian interpolating the iteration of the discrete map defined by the transformations which indeed in numerical applications can be considered a linear combination of the integrals appearing in the spectral curve of the lax matrix an example with the periodic toda lattice is given

sub gaussian estimates of heat kernels on infinite graphs we prove that a two sided sub gaussian estimate of the heat kernel on an infinite weighted graph takes place if and only if the volume growth of the graph is uniformly polynomial and the green kernel admits a uniform polynomial decay

integrable symplectic maps abstract in this paper we first give a terse survey of symplectic maps their canonical formulation and integrability then we introduce a rigorous procedure to construct integrable symplectic maps starting from integrable evolution equations on lattices a number of illustrative examples are provided

on decoupling inequalities and percolation of excursion sets of the gaussian free field we prove decoupling inequalities for the gaussian free field on mathbb z d d ge 3 as an application we obtain exponential decay with logarithmic correction for d 3 of the connectivity function of excursion sets for large values of the threshold

random walks on discrete cylinders with large bases and random interlacements following the recent work of sznitman probab theory related fields 145 2009 143 174 we investigate the microscopic picture induced by a random walk trajectory on a cylinder of the form g n x z where g n is a large finite connected weighted graph and relate it to the model of random interlacements on infinite transient weighted graphs under suitable assumptions the set of points not visited by the random walk until a time of order g n 2 in a neighborhood of a point with z component of order g n converges in distribution to the law of the vacant set of a random interlacement on a certain limit model describing the structure of the graph in the neighborhood of the point the level of the random interlacement depends on the local time of a brownian motion the result also describes the limit behavior of the joint distribution of the local pictures in the neighborhood of several distant points with possibly different limit models as examples of g n we treat the d dimensional box of side length n the sierpinski graph of depth n and the d ary tree of depth n where d 2

growth and integrability in the dynamics of mappings the growth of some numerical characteristics of the mappings under their iterations in the context of the general problem of integrability is discussed in the general case such characteristics as complexity by arnold or the number of the different images for the multiple valued mappings are growing exponentially it is shown that the integrability is closely related with thepolynomial growth the analogies with quantum integrable systems are discussed

peakons r matrix and toda lattice abstract the integrability of a family of hamiltonian systems describing in a particular case the motion of n peakons special solutions of the so called camassa holm equation is established in the framework of the r matrix approach starting from its lax representation in the general case the r matrix is a dynamical one and has an interesting though complicated structure however for a particular choice of the relevant parameters in the hamiltonian the one corresponding to the pure peakons case the r matrix becomes essentially constant and reduces to the one pertaining to the finite non periodic toda lattice intriguing consequences of this property are discussed and an integrable time discretisation is derived

elliptic solutions to difference non linear equations and related many body problems we study algebro geometric finite gap and elliptic solutions of fully discretized kp or 2d toda equations in bilinear form they are hirota s difference equation for functions starting from a given algebraic curve we express the function and the baker akhiezer function in terms of the riemann theta function we show that the elliptic solutions when the function is an elliptic polynomial form a subclass of the general algebro geometric solutions we construct the algebraic curves of the elliptic solutions the evolution of zeros of the elliptic solutions is governed by the discrete time generalization of the ruijsenaars schneider many body system the zeros obey equations which have the form of nested bethe ansatz equations known from integrable quantum field theories we discuss the lax representation and the action angle type variables for the many body system we also discuss elliptic solutions to discrete analogues of kdv sine gordon and 1d toda equations and describe the loci of the zeros

disconnection random walks and random interlacements we consider random interlacements on mathbb z d d ge 3 when their vacant set is in a strongly percolative regime we derive an asymptotic upper bound on the probability that the random interlacements disconnect a box of large side length from the boundary of a larger homothetic box as a corollary we obtain an asymptotic upper bound on a similar quantity where the random interlacements are replaced by the simple random walk it is plausible but open at the moment that these asymptotic upper bounds match the asymptotic lower bounds obtained by xinyi li and the author in electron j probab 19 17 1 26 2014 for random interlacements and by xinyi li in a lower bound on disconnection by simple random walk arxiv 1412 3959 2014 for the simple random walk in any case our bounds capture the principal exponential rate of decay of these probabilities in any dimension d ge 3

on pinned fields interlacements and random walk on mathbb z n mathbb z 2 we define two families of poissonian soups of bidirectional trajectories on mathbb z 2 which can be seen to adequately describe the local picture of the trace left by a random walk on the two dimensional torus mathbb z n mathbb z 2 started from the uniform distribution run up to a time of order n log n 2 and forced to avoid a fixed point the local limit of the latter was recently established in comets et al commun math phys 343 129 164 2016 our construction proceeds by considering somewhat in the spirit of statistical mechanics a sequence of finite volume approximations consisting of random walks avoiding the origin and killed at spatial scale n either using dirichlet boundary conditions or by means of a suitably adjusted mass by tuning the intensity u of such walks with n the occupation field can be seen to have a nontrivial limit corresponding to that of the actual random walk our construction thus yields a two dimensional analogue of the random interlacements model introduced in sznitman ann math 171 3 2039 2087 2010 in the transient case it also links it to the pinned free field in mathbb z 2 by means of a pinned ray knight type isomorphism theorem

a lower bound for disconnection by simple random walk we consider simple random walk on zdzd d 3d 3 motivated by the work of a s sznitman and the author in probab theory related fields 161 2015 309 350 and electron j probab 19 2014 1 26 we investigate the asymptotic behavior of the probability that a large body gets disconnected from infinity by the set of points visited by a simple random walk we derive asymptotic lower bounds that bring into play random interlacements although open at the moment some of the lower bounds we obtain possibly match the asymptotic upper bounds recently obtained in disconnection random walks and random interlacements 2014 this potentially yields special significance to the tilted walks that we use in this work and to the strategy that we employ to implement disconnection

harnack inequalities and sub gaussian estimates for random walks we show that the beta parabolic harnack inequality for random walks on graphs is equivalent on one hand to the sub gaussian estimate for the transition probability and on the other hand to the conjunction of the elliptic harnack inequality the doubling volume property and the fact that the mean exit time in any ball of radius r is of the order r beta the latter condition can be replaced by a certain estimate of the resistance of annuli

the non dynamical r matrix structure for the elliptic calogero moser model in this paper we construct a new lax operator for the elliptic calogero moser model with general from the classical dynamical twisting in which the corresponding r matrix is purely numerical non dynamic the non dynamical r matrix structure of this lax operator is obtained which is the elliptic symmetric r matrix

tracking periodic signals with the overtaking criterion the overtaking criterion is suggested to evaluate the tracking of periodic signals on an infinite horizon existence uniqueness and a feedback form are established the asymptotic dynamics of the solution and its relation to the asymptotic performance of the finite time optimizers are examined

random walks on discrete cylinders and random interlacements we explore some of the connections between the local picture left by the trace of simple random walk on a cylinder mathbb z n mathbb z d times mathbb z d 2 running for times of order n2d and the model of random interlacements recently introduced in sznitman http www math ethz ch u sznitman preprints in particular we show that for large n in the neighborhood of a point of the cylinder with vertical component of order nd the complement of the set of points visited by the walk up to times of order n2d is close in distribution to the law of the vacant set of random interlacements with a level which is determined by an independent brownian local time the limit behavior of the joint distribution of the local pictures in the neighborhood of finitely many points is also derived

percolation in the vacant set of poisson cylinders we consider a poisson point process on the space of lines in mathbb r d where a multiplicative factor u 0 of the intensity measure determines the density of lines each line in the process is taken as the axis of a bi infinite cylinder of radius 1 we investigate percolative properties of the vacant set defined as the subset of mathbb r d that is not covered by any such cylinder we show that in dimensions d 4 there is a critical value u d in 0 infty such that with probability 1 the vacant set has an unbounded component if u u d for d 3 we prove that the vacant set does not percolate for large u and that the vacant set intersected with a two dimensional subspace of mathbb r d does not even percolate for small u 0

boundary conditions for integrable quantum systems a new class of boundary conditions is described for quantum systems integrable by means of the quantum inverse scattering r matrix method the method proposed allows the author to treat open quantum chains with appropriate boundary terms in the hamiltonian the general considerations are applied to the xxz and xyz models the nonlinear schrodinger equation and toda chain

generalized bethe ansatz with the general spin representations of the sklyanin algebra the quantum inverse scattering method is applied to the inhomogeneous heisenberg xyz model with general spins i e the general spin representations of the sklyanin algebra by means of the generalized bethe ansatz developed by takhtajan and faddeev 1979 the eigenvectors and eigenvalues of the transfer matrix are found

new hierarchies of isospectral and non isospectral integrable nlees derived from the harry dym spectral problem abstract by making use of our previous framework we display the hierarchies of generalized nonlinear evolution equations gnlees associated with the harry dym spectral problem and construct the corresponding generalized lax representations glr in this paper it will be clearly seen that the generalized hierarchies can include not only the well known harry dym hierarchy of isospectral nlees but also other new integrable hierarchies of isospectral and non isospectral nlees through choosing the so called seed function these new hierarchies of nlees give some new integrable evolution equations which are very likely to have potential applications in theoretical and experimental physics all of these hierarchies of nlees possess the glr

spin generalization of the calogero moser hierarchy and the matrix kp hierarchy we establish a correspondence between rational solutions to the matrix kp hierarchy and the spin generalization of the calogero moser system on the level of hierarchies namely it is shown that the rational solutions to the matrix kp hierarchy appear to be isomorphic to the spin calogero moser system in a sense that the dynamics of poles of solutions to the matrix kp hierarchy in the higher times is governed by the higher hamiltonians of the spin calogero moser integrable hierarchy with rational potential

on the fragmentation of a torus by random walk we consider a simple random walk on a discrete torus input amssym bbb z n bbb z d with dimension d 3 and large side length n for a fixed constant u 0 we study the percolative properties of the vacant set consisting of the set of vertices not visited by the random walk in its first und steps we prove the existence of two distinct phases of the vacant set in the following sense if u 0 is chosen large enough all components of the vacant set contain no more than log n u vertices with high probability as n tends to infinity on the other hand for small u 0 there exists a macroscopic component of the vacant set occupying a nondegenerate fraction of the total volume nd in dimensions d 5 we additionally prove that this macroscopic component is unique by showing that all other components have volumes of order at most log n u our results thus solve open problems posed by benjamini and sznitman who studied the small u regime in high dimension the proofs are based on a coupling of the random walk with random interlacements on input amssym bbb z d among other techniques the construction of this coupling employs a refined use of discrete potential theory by itself this coupling strengthens a result by windisch 2011 wiley periodicals inc

integrable discretizations of the spin ruijsenaars schneider models integrable discretizations are introduced for the rational and hyperbolic spin ruijsenaars schneider models these discrete dynamical systems are demonstrated to belong to the same integrable hierarchies as their continuous time counterparts explicit solutions are obtained for arbitrary flows of the hierarchies including the discrete time ones

l a b representation for nonlinear evolution equations and its applications in this paper we consider the l a b representation of nonlinear evolution equations nlees and present an approach to determine its range the method we adopt is that of lie quotient algebras as an example we consider the kaup newell spectral problem and give a category of nlees associated with it particularly we obtain the negative order hierarchies of nlees by considering the inverse of recursion operator

quenched invariance principle for simple random walk on clusters in correlated percolation models we prove a quenched invariance principle for simple random walk on the unique infinite percolation cluster for a general class of percolation models on mathbb z d d ge 2 with long range correlations introduced in drewitz et al in j math phys 55 8 083307 2014 solving one of the open problems from there this gives new results for random interlacements in dimension d ge 3 at every level as well as for the vacant set of random interlacements and the level sets of the gaussian free field in the regime of the so called local uniqueness which is believed to coincide with the whole supercritical regime an essential ingredient of our proof is a new isoperimetric inequality for correlated percolation models

on the size of a finite vacant cluster of random interlacements with small intensity in this paper we establish some properties of percolation for the vacant set of random interlacements for d 5 and small intensity u the model of random interlacements was first introduced by sznitman in ann math arxiv 0704 2560 2010 it is known that for small u almost surely there is a unique infinite connected component in the vacant set left by the random interlacements at level u see sidoravicius and sznitman commun pure appl math 62 6 831 858 2009 and teixeira ann appl probab 19 1 454 466 2009 we estimate here the distribution of the diameter and the volume of the vacant component at level u containing the origin given that it is finite this comes as a by product of our main theorem which proves a stretched exponential bound on the probability that the interlacement set separates two macroscopic connected sets in a large cube as another application we show that with high probability the unique infinite connected component of the vacant set is ubiquitous in large neighborhoods of the origin

optimal dynamic output feedback for nonzero set point regulation the discrete time case standard discrete time lqg theory is generalized to a regulation problem involving a priori specified nonzero set points for the state and control variables and nonzero mean disturbances the optimal control law consists of a closed loop component feeding back measurements and a constant open loop component which accounts for the nonzero set point and the nonzero mean disturbance for generality the results are obtained for the problem of fixed order i e full and reduced order dynamic compensation it is shown that the closed loop controller can be designed independently of the open loop control

complete integrability of relativistic calogero moser systems and elliptic function identities poincare invariant generalizations of the galilei invariant calogero moser a particle systems are studied a quantization of the classical integrals s1 sn is presented such that the operators si9 sn mutually commute as a corollary it follows that si9 9sn poisson commute these results hinge on functional equations satisfied by the weierstrass and 0 functions a generalized cauchy identity involving the function leads to an n x n matrix l whose symmetric functions are proportional to sl5 9sn

on the disconnection of a discrete cylinder by a random walk we investigate the large n behavior of the time the simple random walk on the discrete cylinder needs to disconnect the discrete cylinder we show that when d 2 this time is roughly of order n 2 d and comparable to the cover time of the slice but substantially larger than the cover timer of the base by the projection of the walk further we show that by the time disconnection occurs a massive clogging typically takes place in the truncated cylinders of height these mechanisms are in contrast with what happens when d 1

the analogue of the backlund transformation for integrable many body systems canonical transformations analogous to the backlund transformation are discussed explicit formulae are given for many body systems of particles interacting in one dimension

on the critical parameter of interlacement percolation in high dimension the vacant set of random interlacements on d d 3 has nontrivial percolative properties it is known from sznitman ann math 171 2010 2039 2087 sidoravicius and sznitman comm pure appl math 62 2009 831 858 that there is a nondegenerate critical value u such that the vacant set at level u percolates when u u we derive here an asymptotic upper bound on u as d goes to infinity which complements the lower bound from sznitman probab theory related fields to appear our main result shows that u is equivalent to log d for large d and thus has the same principal asymptotic behavior as the critical parameter attached to random interlacements on 2d regular trees which has been explicitly computed in teixeira electron j probab 14 2009 1604 1627

characterization of sub gaussian heat kernel estimates on strongly recurrent graphs sub gaussian estimates for random walks are typical of fractal graphs we characterize them in the strongly recurrent case in terms of resistance estimates only without assuming elliptic harnack inequalities

how universal are asymptotics of disconnection times in discrete cylinders we investigate the disconnection time of a simple random walk in a discrete cylinder with a large finite connected base in a recent article of a dembo and the author it was found that for large n the disconnection time of g n x z has rough order g n 2 when g n z nz d in agreement with a conjecture by i benjamini we show here that this behavior has broad generality when the bases of the discrete cylinders are large connected graphs of uniformly bounded degree

quantum integrable systems and representations of lie algebras in this paper the quantum integrals of the hamiltonian of the quantum many body problem with the interaction potential k sinh2 x sutherland operator are constructed as images of higher casimirs of the lie algebra gl n under a certain homomorphism from the center of u gl n to the algebra of differential operators in n variables a similar construction applied to the affine gl n at the critical level k n defines a correspondence between higher sugawara operators and quantum integrals of the hamiltonian of the quantum many body problem with the potential equal to constant times the weierstrass function this allows one to give a new proof of the olshanetsky perelomov theorem stating that this hamiltonian defines a completely integrable quantum system a new expression is also given for eigenfunctions of the quantum integrals of the sutherland operator as traces of intertwining operators between certain representations of gl n

a super integrable discretization of the calogero model a time discretization that preserves the super integrability of the calogero model is obtained by application of the integrable time discretization of the harmonic oscillator to the projection method for the calogero model with continuous time in particular the difference equations of motion which provide an explicit scheme for time integration are explicitly presented for the two body case numerical results exhibit that the scheme conserves all the 3 conserved quantities of the two body calogero model with a precision of the machine epsilon times the number of iterations

the lax representation for an integrable class of relativistic dynamical systems we exhibit the lax pair for the class of relativistic dynamical systems recently introduced by ruijsenaars and schneider whose equations of motion read whereb is an arbitrary constant and the weierstrass elliptic function

upper bound on the disconnection time of discrete cylinders and random interlacements we study the asymptotic behavior for large n of the disconnection time t n of a simple random walk on the discrete cylinder n d x z when d 2 we explore its connection with the model of random interlacements on d 1 recently introduced in ann math in press and specifically with the percolative properties of the vacant set left by random interlacements as an application we show that in the large n limit the tail of t n n 2d is dominated by the tail of the first time when the supremum over the space variable of the brownian local times reaches a certain critical value as a by product we prove the tightness of the laws of t n n 2d when d 2

the optimal projection equations for static and dynamic output feedback the singular case oblique projections have been shown to arise naturally in both static and dynamic optimal design problems for static controllers an oblique projection was inherent in the early work of levine and athans while for dynamic controllers an oblique projection was developed by hyland and bernstein this note is motivated by the following natural question what is the relationship between the oblique projection arising in optimal static output feedback and the oblique projection arising in optimal fixed order dynamic compensation we show that in nonstrictly proper optimal output feedback there are indeed three distinct oblique projections corresponding to singular measurement noise singular control weighting and reduced compensator order moreover we unify the levine athans and hyland bernstein approaches by rederiving the optimal projection equations for combined static dynamic nonstrictly proper output feedback in a form which clearly illustrates the role of the three projections in characterizing the optimal feedback gains even when the dynamic component of the nonstrictly proper controller is of full order the controller is characterized by four matrix equations which generalize the standard lqg result

time discretization of the spin calogero moser model and the semi discrete matrix kp hierarchy we introduce the discrete time version of the spin calogero moser system the equations of motion follow from the dynamics of poles of rational solutions to the matrix kadomtsev petviashvili hierarchy with discrete time the dynamics of poles is derived using the auxiliary linear problem for the discrete flow

dynamical r matrix for the elliptic ruijsenaars schneider system the classical r matrix structure for the generic elliptic ruijsenaars schneider model is presented it makes manifest the integrability of this model as well as of its discrete time version that was constructed in a recent paper

universal whitham hierarchy dispersionless hirota equations and multicomponent kp hierarchy abstract the goal of this paper is to identify the universal whitham hierarchy of genus zero with a dispersionless limit of the multicomponent kp hierarchy to this end the multicomponent kp hierarchy is re formulated to depend on several discrete variables called charges these discrete variables play the role of lattice coordinates in underlying toda field equations a multicomponent version of the so called differential fay identity are derived from the hirota equations of the function of this charged multicomponent kp hierarchy these multicomponent differential fay identities have a well defined dispersionless limit the dispersionless hirota equations the dispersionless hirota equations turn out to be equivalent to the hamilton jacobi equations for the s functions of the universal whitham hierarchy the differential fay identities themselves are shown to be a generating functional expression of auxiliary linear equations for scalar valued wave functions of the multicomponent kp hierarchy

gumbel fluctuations for cover times in the discrete torus this work proves that the fluctuations of the cover time of simple random walk in the discrete torus of dimension at least three with large side length are governed by the gumbel extreme value distribution this result was conjectured for example in aldous and fill reversible markov chains and random walks on graphs in preparation we also derive some corollaries which qualitatively describe how covering happens in addition we develop a new and stronger coupling of the model of random interlacements introduced by sznitman ann math 2 171 3 2039 2087 2010 and random walk in the torus this coupling is used to prove the cover time result and is also of independent interest

on chemical distances and shape theorems in percolation models with long range correlations in this paper we provide general conditions on a one parameter family of random infinite subsets of z d to contain a unique infinite connected component for which the chemical distances are comparable to the euclidean distance in addition we show that these conditions also imply a shape theorem for the corresponding infinite connected component by verifying these conditions for specific models we obtain novel results about the structure of the infinite connected component of the vacant set of random interlacements and the level sets of the gaussian free field as a byproduct we obtain alternative proofs to the corresponding results for random interlacements in the work of cern and popov on the internal distance in the interlacement set electron j probab 17 29 1 25 2012 and while our main interest is in percolation models with long range correlations we also recover results in the spirit of the work of antal and pisztora on the chemical distance for supercritical bernoulli percolation ann probab 24 2 1036 1048 1996 for bernoulli percolation finally as a corollary we derive new results about the chemical diameter of the largest connected component in the complement of the trace of the random walk on the torus

the optimal projection equations for fixed order dynamic compensation first order necessary conditions for quadratically optimal steady state fixed order dynamic compensation of a linear time invariant plant in the presence of disturbance and observation noise are derived in a new and highly simplified form in contrast to the pair of matrix riccati equations for the full order lqg case the optimal steady state fixed order dynamic compensator is characterized by four matrix equations two modified riccati equations and two modified lyapunov equations coupled by a projection whose rank is precisely equal to the order of the compensator and which determines the optimal compensator gains the coupling represents a graphic portrayal of the demise of the classical separation principle for the reduced order controller case

phase transition in loop percolation we are interested in the clusters formed by a poisson ensemble of markovian loops on infinite graphs this model was introduced and studied in le jan c r math acad sci paris 350 13 14 643 646 2012 ill j math 57 2 525 558 2013 it is a model with long range correlations with two parameters alpha and kappa the non negative parameter alpha measures the amount of loops and kappa plays the role of killing on vertices penalizing kappa 0 or favoring kappa 0 appearance of large loops it was shown in le jan ill j math 57 2 525 558 2013 that for any fixed kappa and large enough alpha there exists an infinite cluster in the loop percolation on mathbb z d in the present article we show a non trivial phase transition on the integer lattice mathbb z d d ge 3 for kappa 0 more precisely we show that there is no loop percolation for kappa 0 and alpha small enough interestingly we observe a critical like behavior on the whole sub critical domain of alpha namely for kappa 0 and any sub critical value of alpha the probability of one arm event decays at most polynomially for d ge 5 we prove that there exists a non trivial threshold for the finiteness of the expected cluster size for alpha below this threshold we calculate up to a constant factor the decay of the probability of one arm event two point function and the tail distribution of the cluster size these rates are comparable with the ones obtained from a single large loop and only depend on the dimension for d 3 or 4 we give better lower bounds on the decay of the probability of one arm event which show importance of small loops for long connections in addition we show that the one arm exponent in dimension 3 depends on the intensity alpha

integrable time discretisation of the ruijsenaars schneider model an exactly integrable symplectic correspondence is derived which in a continuum limit leads to the equations of motion of the relativistic generalization of the calogero moser system that was introduced for the first time by ruijsenaars and schneider for the discrete time model the equations of motion take the form of bethe ansatz equations for the inhomogeneous spin 1 2 xyz heisenberg magnet we present a lax pair the sympletic structure and prove the involutivity of the invariants exact solutions are investigated in the rational and hyperbolic trigonometric limits of the system that is given in terms of elliptic functions these solutions are connected with discrete soliton equations the results obtained allow us to consider the bethe ansatz equations as ones giving an integrable symplectic correspondence mixing the parameters of the quantum integrable system and the parameters of the corresponding bethe wavefunction

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

vacant set of random interlacements and percolation we introduce a model of random interlacements made of a countable collection of doubly infinite paths on d d 3 a nonnegative parameter u measures how many trajectories enter the picture this model describes in the large n limit the microscopic structure in the bulk which arises when considering the disconnection time of a discrete cylinder n d 1 x by simple random walk or the set of points visited by simple random walk on the discrete torus n d at times of order u n d in particular we study the percolative properties of the vacant set left by the interlacement at level u which is an infinite connected translation invariant random subset of d we introduce a critical value u such that the vacant set percolates for u u our main results show that u is finite when d 3 and strictly positive when d 7

percolation for the vacant set of random interlacements we investigate random interlacements on d d 3 this model recently introduced in 8 corresponds to a poisson cloud on the space of doubly infinite trajectories modulo time shift tending to infinity at positive and negative infinite times a nonnegative parameter u measures how many trajectories enter the picture our main interest lies in the percolative properties of the vacant set left by random interlacements at level u we show that for all d 3 the vacant set at level u percolates when u is small this solves an open problem of 8 where this fact has only been established when d 7 it also completes the proof of the nondegeneracy in all dimensions d 3 of the critical parameter u of 8 2008 wiley periodicals inc

integrable discretizations of the bogoyavlensky lattices discretizations of the bogoyavlensky lattices are introduced belonging to the same hierarchies as the continuous time systems the construction exemplifies the general scheme for integrable discretization of systems on lie algebras with r matrix poisson brackets an initial value problem for the difference equations is solved in terms of a factorization problem in a group interpolating hamiltonian flows are found

quantum integrable systems and elliptic solutions of classical discrete nonlinear equations in spite of the diversity of solvable models of quantum field theory and the vast variety of methods the final results display dramatic unification the spectrum of an integrable theory with a local interaction is given by a sum of elementary energies e sum limits i varepsilon left u i right 1 1 where u i obey a system of algebraic or transcendental equations known as bethe equations 1 2 the major ingredients of bethe equations are determined by the algebraic structure of the problem a typical example of a system of bethe equations related to a i type models with elliptic r matrix is e 4 eta nu frac phi left u j right phi left u j 2 right mathop pi limits k frac sigma left eta left u j u k 2 right right sigma left eta left u j u k 2 right right 1 2 where x is the weierstrass function and phi left u right mathop pi limits k 1 n sigma left eta left u y k right right 1 3

a generalisation of the calogero moser system abstract a generalised calogero moser many body system in which the particles possess extra internal degrees of freedom is introduced and solved by the method of olshanetsky and perelomov further the hierarchy of flows commuting with this system is obtained from a hierarchy of linear systems possessing sl n symmetry by the usual method of reduction thus they are shown to be integrable

decoupling inequalities and interlacement percolation on g z we study the percolative properties of random interlacements on g where g is a weighted graph satisfying certain sub gaussian estimates attached to the parameters 1 and 2 1 describing the respective polynomial growths of the volume on g and of the time needed by the walk on g to move to a distance we develop decoupling inequalities which are a key tool in showing that the critical level u for the percolation of the vacant set of random interlacements is always finite in our set up and that it is positive when 1 2 we also obtain several stretched exponential controls both in the percolative and non percolative phases of the model even in the case where g d d 2 several of these results are new

tetrahedra octahedra and cubo octahedra integrable geometry of multi ratios geometric and algebraic aspects of multi ratios m2n are investigated in detail connections with menelaus theorem clifford configurations and maxwell s reciprocal quadrangles are utilized to associate the multi ratios m4 m8 and m8 with tetrahedra octahedra and cubo octahedra respectively integrable maps defined on face centred fcc lattices and irregular lattices composed of the face centres of simple cubic lattices are constructed and related to the discrete kp and bkp equations and the integrable discrete darboux system governing conjugate lattices an interpretation in terms of integrable irregular lattices of slopes on the plane is also given

kp hierarchy for the cyclic quiver we introduce a generalisation of the kp hierarchy closely related to the cyclic quiver and the cherednik algebra hk zm this hierarchy depends on m parameters one of which can be eliminated with the usual kp hierarchy corresponding to the m 1 case generalising the result of wilson invent math 133 1 1 41 1998 we show that our hierarchy admits solutions parameterised by suitable quiver varieties the pole dynamics for these solutions is shown to be governed by the classical calogero moser system for the wreath product zm sn and its new spin version these results are further extended to the case of the multi component hierarchy