wave propagation on riemannian symmetric spaces abstract let x g k be a riemannian symmetric space of the noncompact type and let l x be the laplace beltrami operator on x we consider on x r a differential equation of the type l x u p t u where p t is a second order differential operator in t r using the radon transform on x we relate huygens principle for the solutions u to this equation to the corresponding question for the solutions v to the equation l a v p t r v on a maximal flat subspace a where r is a certain explicit constant in particular we conclude that huygens principle holds for solutions to the modified wave equation on x obtained by letting p t t 2 r when x is odd dimensional and g has one conjugacy class of cartan subgroups

on the continuity of the multivariate pade operator abstract continuity of the univariate pade operator was proved in 5 6 we discuss the limitations of a multivariate generalization and prove a multivariate analogon of the continuity property

multivariate pade approximation abstract this paper is a survey on the multivariate pade approximation two types of approximants are considered those which can approximate general meromorphic functions f h g where both h and g are holomorphic and those which are specialized to the approximation of functions of the same form where g is a polynomial algorithms are described together with the different techniques used for proving convergence

singular rules for the calculations of non normal multivariate pade approximants abstract section 1 describes the univariate situation in the case of non normal pade approximants and cordellier s extension of the famous five star identity of wynn section 2 repeats our definition of multivariate pade approximants and proves a number of theorems that remain valid when going from the univariate to the multivariate case these theorems and more new results given in section 3 will finally also copy cordellier s extension from the univariate to the multivariate case

the logarithmic sobolev inequality for discrete spin systems on a lattice for finite range lattice gases with a finite spin space it is shown that the dobrushin shlosman mixing condition is equivalent to the existence of a logarithmic sobolev inequality for the associated unique gibbs state in addition implications of these considerations for the ergodic properties of the corresponding glauber dynamics are examined

the lee yang and polya schur programs i linear operators preserving stability in 1952 lee and yang proposed the program of analyzing phase transitions in terms of zeros of partition functions linear operators preserving non vanishing properties are essential in this program and various contexts in complex analysis probability theory combinatorics and matrix theory we characterize all linear operators on finite or infinite dimensional spaces of multivariate polynomials preserving the property of being non vanishing whenever the variables are in prescribed open circular domains in particular this solves the higher dimensional counterpart of a long standing classification problem originating from classical works of hermite laguerre hurwitz and polya schur on univariate polynomials with such properties

explicit construction of multivariate pade approximants we explicitly construct both homogeneous and nonhomogeneous multivariate pade approximants to some functions which satisfy some simple functional equations by using the residue theorem and the functional equation method which has been used successfully by borwein 1988 to construct one variable pade approximants to the q elementary functions

equipartition of energy for waves in symmetric space abstract let x g k be an odd dimensional semisimple riemannian symmetric space of the noncompact type and suppose that all cartan subgroups of g are conjugate let u be a real valued classical solution of the modified wave equation utt k u on r x the cauchy data of which are supported in a closed metric ball of radius a at time t 0 here t is the coordinate on r is the nonpositive definite laplace beltrami operator on x and k is a positive constant depending on the root structure of the lie algebra of g we show that the t independent energy functional of u is eventually for t a partitioned into equal potential and kinetic parts specifically half the integrals over x of ut2 and du 2 ku 2 respectively where d is the exterior derivative in x the proof uses helgason s paley wiener theorem for x the classical paley wiener theorem and properties of harish chandra s c function

algorithmic pirogov sinai theory we develop an efficient algorithmic approach for approximate counting and sampling in the low temperature regime of a broad class of statistical physics models on finite subsets of the lattice mathbb z d and on the torus mathbb z n mathbb z d our approach is based on combining contour representations from pirogov sinai theory with barvinok s approach to approximate counting using truncated taylor series some consequences of our main results include an fptas for approximating the partition function of the hard core model at sufficiently high fugacity on subsets of mathbb z d with appropriate boundary conditions and an efficient sampling algorithm for the ferromagnetic potts model on the discrete torus mathbb z n mathbb z d at sufficiently low temperature

the repulsive lattice gas the independent set polynomial and the lovasz local lemma we elucidate the close connection between the repulsive lattice gas in equi librium statistical mechanics and the lovasz local lemma in probabilistic com binatorics we show that the conclusion of the lovasz local lemma holds for dependency graph g and probabilities px if and only if the independent set polynomial for g is nonvanishing in the polydisc of radii px furthermore we show that the usual proof of the lovasz local lemma which provides a suffi cient condition for this to occur corresponds to a simple inductive argument for the nonvanishing of the independent set polynomial in a polydisc which was discovered implicitly by shearer 98 and explicitly by dobrushin 37 38 we also present some refinements and extensions of both arguments including a generalization of the lovasz local lemma that allows for soft dependencies in addition we prove some general properties of the partition function of a repulsive lattice gas most of which are consequences of the alternating sign property for the mayer coefficients we conclude with a brief discussion of the repulsive lattice gas on countably infinite graphs

integrability and huygens principle on symmetric spaces the explicit formulas for fundamental solutions of the modified wave equations on certain symmetric spaces are found these symmetric spaces have the following characteristic property all multiplicities of their restricted roots are even as a corollary in the odd dimensional case one has that the huygens principle in hadamard s sense for these equations is fulfilled we consider also the heat and laplace equations on such a symmetric space and give explicitly the corresponding fundamental solutions heat kernel and green s function this continues our previous investigations 16 of the spherical functions on the same symmetric spaces based on the fact that the radial part of the laplace beltrami operator on such a space is related to the algebraically integrable case of the generalised calogero sutherland moser quantum system in the last section of this paper we apply the methods of heckman and opdam to extend our results to some other symmetric spaces in particular to complex and quaternian grassmannians

quasi factorization of the entropy and logarithmic sobolev inequalities for gibbs random fields we show that the entropy functional exhibits a quasi factorization property with respect to a pair of weakly dependent algebras as an application we give a simple proof that the dobrushin and shlosmans complete analyticity condition for a gibbs specification with finite range summable interaction implies uniform logarithmic sobolev inequalities this result has been previously proven using several different techniques the advantage of our approach is that it relies almost entirely on a general property of the entropy while very little is assumed on the dirichlet form no topology is introduced on the single spin space thus discrete and continuous spins can be treated in the same way

the qd algorithm and multivariate pade approximants the quotient difference qd algorithm can be used to construct univariate pade approximants 1 in this paper we see that it can also be used to construct the multivariate pade approximants introduced in 3 just by reformulating the quotient difference algorithm as in sect 1 the multivariate pade approximants and the multivariate qd scheme are treated in the sects 2 and 3 respectively thus for this type of multivariate pade approximants a link with the theory of multivariate continued fractions is established

huygens principle for wave equations on symmetric spaces let x gk be a symmetric space lx the laplace betrami operator on x and 2 the sum of the positive restricted roots of x with multiplicity using the fourier transform on x we write down the solution of the modified wave equation 2u t2 lx 2 u on x it is proved using the paley wiener theorem for this fourier transform s helgason ann of math 98 1973 451 480 that the equation satisfies huygens principle if dim x is odd and all cartan subgroups of g are conjugate

approach to equilibrium of glauber dynamics in the one phase region i the attractive case various finite volume mixing conditions in classical statistical mechanics are reviewed and critically analyzed in particular somefinite size conditions are discussed together with their implications for the gibbs measures and for the approach to equilibrium of glauber dynamics inarbitrarily large volumes it is shown that dobrushin shlosman s theory ofcomplete analyticity and its dynamical counterpart due to stroock and zegarlinski cannot be applied in general to the whole one phase region since it requires mixing properties for regions ofarbitrary shape an alternative approach based on previous ideas of oliveri and picco is developed which allows to establish results on rapid approach to equilibrium deeply inside the one phase region in particular in the ferromagnetic case we considerably improve some previous results by holley and aizenman and holley our results are optimal in the sene that for example they show for the first time fast convergence of the dynamicsfor any temperature above the critical one for thed dimensional ising model with or without an external field in part ii we extensively consider the general case not necessarily attractive and we develop a new method based on renormalizations group ideas and on an assumption of strong mixing in a finite cube to prove hypercontractivity of the markov semigroup of the glauber dynamics

identification of discontinuous nonlinear systems via a multivariate pade approach we present a nonlinear system identification technique based on multi dimensional rational polynomials a multi dimensional pade legendre approximation is developed to circumvent challenges in dealing with sharp shocks the purpose of this paper is to investigate the accuracy of such approximations for identification of various nonlinear systems particularly systems with a non smooth response surface this identification approach utilizes the generalized form of a pade legendre approximation for studying multivariable functions in the studied problems the nonlinearity is a function of state variables displacement and velocity which requires multi dimensional formulation furthermore a spatial filter is applied to minimize the effects of the singular points in the applicable rational function of the response surface this study presents different types of nonlinearities including smooth irregular and hysteretic functions in order to demonstrate the performance of the approach under different conditions in order to study the robustness of the method in comparison to other identification techniques based on plain polynomial representation a nonlinear system with a sharp discontinuous restoring force surface is considered the performance of both approaches is investigated for different degrees of sharpness in addition the accuracy of the identified models to represent the nonlinear system is verified by comparing the output of the system computed on the basis of the identified model from data sets corresponding to different excitations than those used for identification purposes it is shown that the proposed approach provides a robust identification technique for a broad class of highly nonlinear systems and it is particularly advantageous to use when dealing with systems incorporating discontinuous properties

improved bounds for sampling colorings we consider the problem of sampling uniformly at random from the set of proper k colorings of a graph with maximum degree our main result is the design of a simple markov chain that converges in o nk log n time to the desired distribution when k 116

huygens principle and integrable systems abstract the theory of integrable systems in particular of the generalised calogero moser problem is applied to constructing and investigating examples of the hyperbolic equations satisfying huygens principle in hadamard s sense

glauber dynamics on trees and hyperbolic graphs we study continuous time glauber dynamics for random configurations with local constraints e g proper coloring ising and potts models on finite graphs with n vertices and of bounded degree we show that the relaxation time defined as the reciprocal of the spectral gap 1 2 for the dynamics on trees and on planar hyperbolic graphs is polynomial in n for these hyperbolic graphs this yields a general polynomial sampling algorithm for random configurations we then show that for general graphs if the relaxation time 2 satisfies 2 o 1 then the correlation coefficient and the mutual information between any local function which depends only on the configuration in a fixed window and the boundary conditions decays exponentially in the distance between the window and the boundary for the ising model on a regular tree this condition is sharp

a mean field spin glass with short range interactions we formulate and study a spin glass model on the bethe lattice appropriate boundary fields replace the traditional self consistent methods they give our model well defined thermodynamic properties we establish that there is a spin glass transition temperature above which the single site magnetizations vanish and below which the edwards anderson order parameter is strictly positive in a neighborhood below the transition temperature we use bifurcation theory to establish the existence of a nontrivial distribution of single site magnetizations two properties of this distribution are studied the leading perturbative correction to the gaussian scaling form at the transition and the nonperturbative behavior of the tails

limiting measure of lee yang zeros for the cayley tree this paper is devoted to an in depth study of the limiting measure of lee yang zeroes for the ising model on the cayley tree we build on previous works of muller hartmann and zittartz z phys b 22 59 1975 muller hartmann z phys b 27 161 168 1977 barata and marchetti j stat phys 88 231 268 1997 and barata and goldbaum j stat phys 103 857 891 2001 to determine the support of the limiting measure prove that the limiting measure is not absolutely continuous with respect to lebesgue measure and determine the pointwise dimension of the measure at lebesgue a e point on the unit circle and every temperature the latter is related to the critical exponents for the phase transitions in the model as one crosses the unit circle at lebesgue a e point providing a global version of the phase transition of continuous order discovered by muller hartmann zittartz the key techniques are from dynamical systems because there is an explicit formula for the lee yang zeros of the finite cayley tree of level n in terms of the n th iterate of an expanding blaschke product a subtlety arises because the conjugacies between blaschke products at different parameter values are not absolutely continuous

the lee yang and polya schur programs ii theory of stable polynomials and applications in the first part of this series we characterized all linear operators on spaces of multivariate polynomials preserving the property of being nonvanishing in products of open circular domains for such sets this completes the multivariate generalization of the classification program initiated by polya and schur for univariate real polynomials we build on these classification theorems to develop here a theory of multivariate stable polynomials applications and examples show that this theory provides a natural framework for dealing in a uniform way with lee yang type problems in statistical mechanics combinatorics and geometric function theory in one or several variables in particular we answer a question of hinkkanen on multivariate apolarity

the ising partition function zeros and deterministic approximation we study the problem of approximating the partition function of the ferromagnetic ising model with both pairwise as well as higher order interactions equivalently in graphs as well as hypergraphs our approach is based on the classical lee yang theory of phase transitions along with a new lee yang theorem for the ising model with higher order interactions and on an extension of ideas developed recently by barvinok and patel and regts that can be seen as an algorithmic realization of the lee yang theory our first result is a deterministic polynomial time approximation scheme an fptas for the partition function in bounded degree graphs that is valid over the entire range of parameters beta the interaction and lambda the external field except for the case left lambda right 1 the zero field case a polynomial time randomized approximation scheme fpras for all graphs and all beta lambda based on markov chain monte carlo simulation has long been known unlike most other deterministic approximation algorithms for problems in statistical physics and counting our algorithm does not rely on the decay of correlations property but as pointed out above on lee yang theory this approach extends to the more general setting of the ising model on hypergraphs of bounded degree and edge size where no previous algorithms even randomized were known for a wide range of parameters in order to achieve this latter extension we establish a tight version of the lee yang theorem for the ising model on hypergraphs improving a classical result of suzuki and fisher

glauber dynamics on trees boundary conditions and mixing time we give the first comprehensive analysis of the effect of boundary conditions on the mixing time of the glauber dynamics in the so called bethe approximation specifically we show that the spectral gap and the log sobolev constant of the glauber dynamics for the ising model on an n vertex regular tree with boundary are bounded below by a constant independent of n at all temperatures and all external fields this implies that the mixing time is o logn in contrast to the free boundary case where it is not bounded by any fixed polynomial at low temperatures in addition our methods yield simpler proofs and stronger results for the spectral gap and log sobolev constant in the regime where the mixing time is insensitive to the boundary condition our techniques also apply to a much wider class of models including those with hard core constraints like the antiferromagnetic potts model at zero temperature proper colorings and the hard core lattice gas independent sets

spectral gap and logarithmic sobolev inequality for kawasaki and glauber dynamics we prove that the spectral gap of the kawasaki dynamics shrink at the rate of 1 l2 for cubes of sizel provided that some mixing conditions are satisfied we also prove that the logarithmic sobolev inequality for the glauber dynamics in standard cubes holds uniformly in the size of the cube if the dobrushin shlosman mixing condition holds for standard cubes

partition function zeros for aperiodic systems the study of zeros of partition functions initiated by yang and lee provides an important qualitative and quantitative tool in the study of critical phenomena this has frequently been used for periodic as well as hierarchical lattices here we consider magnetic field and temperature zeros of ising model partition functions on several aperiodic structures in 1d we analyze aperiodic chains obtained from substitution rules the most prominent example being the fibonacci chain in 2d we focus on the tenfold symmetric triangular tiling which allows efficient numerical treatment by means of corner transfer matrices

the random cluster model on a homogeneous tree the random cluster model on a homogeneous tree is defined and studied it is shown that for 1 q 2 the percolation probability in the maximal random cluster measure is continuous inp while forq 2 it has a discontinuity at the critical valuep p c q it is also shown that forq 2 there is nonuniqueness of random cluster measures for an entire interval of values ofp the latter result is in sharp contrast to what happens on the integer lattice z d

fractal structure of zeros in hierarchical models we consider an ising and aq state potts model on a diamond hierarchical lattice we give pictures of the distribution of zeros of the partition function in the complex plane of temperatures for several choices ofq these zeros are just the julia set corresponding to the renormalization group transformation

rigidity of the interface in percolation and random cluster models we study conditioned random cluster measures with edge parameter p and cluster weighting factor q satisfying q 1 the conditioning corresponds to mixed boundary conditions for a spin model interfaces may be defined in the sense of dobrushin and these are proved to be rigid in the thermodynamic limit in three dimensions and for sufficiently large values of p this implies the existence of non translation invariant conditioned random cluster measures in three dimensions the results are valid in the special case q 1 thus indicating a property of three dimensional percolation not previously noted

on the ergodic properties of glauber dynamics we show that if there is an infinite volume gibbs measure which satisfies a logarithmic sobolev inequality with local coefficients of moderate growth then the corresponding stochastic dynamics decays to equilibrium exponentially fast in the uniform norm

a complexity dichotomy for partition functions with mixed signs partition functions also known as homomorphism functions form a rich family of graph invariants that contain combinatorial invariants such as the number of k colorings or the number of independent sets of a graph and also the partition functions of certain spin glass models of statistical physics such as the ising model building on earlier work by dyer and greenhill random structures algorithms 17 2000 pp 260 289 and bulatov and grohe theoret comput sci 348 2005 pp 148 186 we completely classify the computational complexity of partition functions our main result is a dichotomy theorem stating that every partition function is either computable in polynomial time or p complete partition functions are described by symmetric matrices with real entries and we prove that it is decidable in polynomial time in terms of the matrix whether a given partition function is in polynomial time or p complete while in general it is very complicated to give an explicit algebraic or combinatorial description of the tractable cases for partition functions described by hadamard matrices these turn out to be central in our proofs we obtain a simple algebraic tractability criterion which says that the tractable cases are those representable by a quadratic polynomial over the field mathbb f 2

support theorem for the fundamental solution to the schrodinger equation on certain compact symmetric spaces abstract in this paper we construct the fundamental solution to the schodinger equation on a compact symmetric space with even root multiplicities using shift operators of heckman and opdam next we prove that the support of the fundamental solution becomes a lower dimensional subset at a rational time whereas its support and its singular support coincide with the whole symmetric space at an irrational time moreover we also show that generalized gauss sums appear in the expression of the fundamental solution

how well can the concept of pade approximant be generalized to the multivariate case what we know about multivariate pade approximation has been developed in the last 25 years in the next sections we compare and discuss many of these results it will become clear that simple properties or requirements such as the uniqueness of the pade approximant and consequently its consistency property can play a crucial role in the development of the multivariate theory a separate section is devoted to a discussion of the convergence properties at the end we include an extensive reference list on the topic

an adaptive strategy for the bivariate solution of finite element problems using multivariate nested pade approximants most engineering applications involving solutions by numerical methods are dependent on several parameters whose impact on the solution may significantly vary from one to the other at times an ev

inapproximability of the tutte polynomial the tutte polynomial of a graph g is a two variable polynomial t g x y that encodes many interesting properties of the graph we study the complexity of the following problem for rationals x and y take as input a graph g and output a value which is a good approximation to t g x y jaeger et al have completely mapped the complexity of exactly computing the tutte polynomial they have shown that this is p hard except along the hyperbola x 1 y 1 1 and at four special points we are interested in determining for which points x y there is a fully polynomial randomised approximation scheme fpras for t g x y under the assumption rp np we prove that there is no fpras at x y if x y is in one of the half planes x np there is no fpras at the point x y 0 email protected when l 2 is a positive integer thus there is no fpras for counting nowhere zero l flows for l 2 this is an interesting consequence of our work since the corresponding decision problem is in p for example for l 6 although our main concern is to distinguish regions of the tutte plane that admit an fpras from those that do not we also note that the latter regions exhibit different levels of intractability at certain points x y for example the integer points on the x axis or any point in the positive quadrant there is a randomised approximation scheme for t g x y that runs in polynomial time using an oracle for an np predicate on the other hand we identify a region of points x y at which even approximating t g x y is as hard as p

lee yang theorems and the complexity of computing averages we study the complexity of computing average quantities related to spin systems such as the mean magnetization and susceptibility in the ferromagnetic ising model and the average dimer count or average size of a matching in the monomer dimer model by establishing connections between the complexity of computing these averages and the location of the complex zeros of the partition function we show that these averages are p hard to compute and hence under standard assumptions computationally intractable in the case of the ising model our approach requires us to prove an extension of the famous lee yang theorem from the 1950s

pade legendre approximants for uncertainty analysis with discontinuous response surfaces a novel uncertainty propagation method for problems characterized by highly non linear or discontinuous system responses is presented the approach is based on a pade legendre pl formalism which does not require modifications to existing computational tools non intrusive approach and it is a global method the paper presents a novel pl method for problems in multiple dimensions which is non trivial in the pade literature in addition a filtering procedure is developed in order to minimize the errors introduced in the approximation close to the discontinuities the numerical examples include fluid dynamic problems characterized by shock waves a simple dual throat nozzle problem with uncertain initial state and the turbulent transonic flow over a transonic airfoil where the flight conditions are assumed to be uncertain results are presented in terms of statistics of both shock position and strength and are compared to monte carlo simulations

phase transitions on nonamenable graphs we survey known results about phase transitions in various models of statistical physics when the underlying space is a nonamenable graph most attention is devoted to transitive graphs and trees

approximating the permanent a randomised approximation scheme for the permanent of a 0 1s presented the task of estimating a permanent is reduced to that of almost uniformly generating perfect matchings in a graph the latter is accomplished by simulating a markov chain whose states are the matchings in the graph for a wide class of 0 1 matrices the approximation scheme is fully polynomial i e runs in time polynomial in the size of the matrix and a parameter that controls the accuracy of the output this class includes all dense matrices those that contain sufficiently many 1 s and almost all sparse matrices in some reasonable probabilistic model for 0 1 matrices of given density for the approach sketched above to be computationally efficient the markov chain must be rapidly mixing informally it must converge in a short time to its stationary distribution a major portion of the paper is devoted to demonstrating that the matchings chain is rapidly mixing apparently the first such result for a markov chain with genuinely c

interlacing families ii mixed characteristic polynomials and the kadison singer problem we use the method of interlacing polynomials introduced in our previous article to prove two theorems known to imply a positive solution to the kadison singer problem the rst is weaver s conjecture ks 2 which is known to imply kadison singer via a projection paving conjecture of akemann and anderson the second is a formulation due to casazza et al of anderson s original paving conjecture s for which we are able to compute explicit paving bounds the proof involves an analysis of the largest roots of a family of polynomials that we call the mixed characteristic polynomials of a collection of matrices

counting in two spin models on d regular graphs supported in part by alfred p sloan research fellowship supported in part by department of defense ndseg fellowships

approximation algorithms for two state anti ferromagnetic spin systems on bounded degree graphs we show that for the anti ferromagnetic ising model on the bethe lattice weak spatial mixing implies strong spatial mixing as a by product of our analysis we obtain what is to the best of our knowledge the first rigorous proof of the uniqueness threshold for the anti ferromagnetic ising model with non zero external field on the bethe lattice following a method due to weitz 15 we then use the equivalence between weak and strong spatial mixing to give a deterministic fully polynomial time approximation scheme for the partition function of the anti ferromagnetic ising model with arbitrary field on graphs of degree at most d throughout the uniqueness region of the gibbs measure on the infinite d regular tree by a standard correspondence our results translate to arbitrary two state anti ferromagnetic spin systems with soft constraints subsequent to a preliminary version of this paper sly and sun 13 have shown that our results are optimal in the sense that under standard complexity theoretic assumptions there does not exist a fully polynomial time approximation scheme for the partition function of such spin systems on graphs of maximum degree d for parameters outside the uniqueness region taken together the results of 13 and of this paper therefore indicate a tight relationship between complexity theory and phase transition phenomena in two state anti ferromagnetic spin systems

griffiths singularities in diluted ising models on the cayley tree the griffiths singularities are fully exhibited for a class of diluted ferromagnetic ising models defined on the cayley tree bethe lattice for the deterministic model the lee yang circle theorem is explicitly proven for the magnetization at the origin and it is shown that in the thermodynamic limit the lee yang singularities become dense in the entire unit circle for the whole ferromagnetic phase smoothness infinite differentiability of the quenched magnetizationm at the origin with respect to the external magnetic field is also proven for convenient choices of temperature and disorder from our analysis we also conclude that the existence of metastable states is impossible for the random models under consideration

random generation of combinatorial structures from a uniform distribution abstract the class of problems involving the random generation of combinatorial structures from a uniform distribution is considered uniform generation problems are in computational difficulty intermediate between classical existence and counting problems it is shown that exactly uniform generation of efficiently verifiable combinatorial structures is reducible to approximate counting and hence is within the third level of the polynomial hierarchy natural combinatorial problems are presented which exhibit complexity gaps between their existence and generation and between their generation and counting versions it is further shown that for self reducible problems almost uniform generation and randomized approximate counting are inter reducible and hence of similar complexity

nested multivariate pade approximants abstract a new class of multivariate pade approximants is introduced when dealing with two variables x and y the approach consists in applying the pade approximation with respect to y to the coefficients of the pade approximation with respect to x this technique has a natural extension to n variables and is well adapted to programming in any usual language the advantage of the method is that the algorithm uses only univariate pade approximation as a consequence only small systems need to be solved and the computation of the approximant is much faster if m is the number of derivatives with respect to x and y the algorithm requires o m 4 operations instead of o m 6 when matrix methods are used for solving the coefficients problem or o m 3 operations instead of o m 4 if staircase methods are used another important feature is that the reliable methods which have been developed for choosing the degrees of the numerator and the denominator in the univariate case can be straightforwardly used in the multivariate case

polynomial time approximation algorithms for the ising model the paper presents a randomised algorithm which evaluates the partition function of an arbitrary ferromagnetic ising system to any specified degree of accuracy the running time of the algorithm in

exact thresholds for ising gibbs samplers on general graphs we establish tight results for rapid mixing of gibbs samplers for the ferromagnetic ising model on general graphs we show that if d 1 tanh 1 then there exists a constant c such that the discrete time mixing time of gibbs samplers for the ferromagnetic ising model on any graph of n vertices and maximal degree d where all interactions are bounded by and arbitrary external fields are bounded by cnlogn moreover the spectral gap is uniformly bounded away from 0 for all such graphs as well as for infinite graphs of maximal degree d we further show that when dtanh 1 with high probability over the erd s renyi random graph g n d n it holds that the mixing time of gibbs samplers is n1 1 loglogn both results are tight as it is known that the mixing time for random regular and erd s renyi random graphs is with high probability exponential in n when d 1 tanh 1 and dtanh 1 respectively to our knowledge our results give the first tight sufficient conditions for rapid mixing of spin systems on general graphs moreover our results are the first rigorous results establishing exact thresholds for dynamics on random graphs in terms of spatial thresholds on trees

fisher zeros and correlation decay in the ising model we study the complex zeros of the partition function of the ising model viewed as a polynomial in the interaction parameter these are known as fisher zeros in light of their introduction by fisher in 1965 fisher m e the nature of critical points in lecture notes in theoretical physics edited by brittin w e university of colorado press 1965 vol 7c while the zeros of the partition function as a polynomial in the field parameter have been extensively studied since the classical work of lee and yang comparatively little is known about fisher zeros for general graphs our main result shows that the zero field ising model has no fisher zeros in a complex neighborhood of the entire region of parameters where the model exhibits correlation decay in addition to shedding light on fisher zeros themselves this result also establishes a formal connection between two distinct notions of phase transition for the ising model the absence of complex zeros analyticity of the free energy density or the normalized logarithm of the partition function and decay of correlations with distance we also discuss the consequences of our result for efficient deterministic approximation of the partition function our proof relies heavily on algorithmic techniques notably weitz s self avoiding walk tree and as such belongs to a growing body of work that uses algorithmic methods to resolve classical questions in statistical physics we study the complex zeros of the partition function of the ising model viewed as a polynomial in the interaction parameter these are known as fisher zeros in light of their introduction by fisher in 1965 fisher m e the nature of critical points in lecture notes in theoretical physics edited by brittin w e university of colorado press 1965 vol 7c while the zeros of the partition function as a polynomial in the field parameter have been extensively studied since the classical work of lee and yang comparatively little is known about fisher zeros for general graphs our main result shows that the zero field ising model has no fisher zeros in a complex neighborhood of the entire region of parameters where the model exhibits correlation decay in addition to shedding light on fisher zeros themselves this result also establishes a formal connection between two distinct notions of phase transition for the ising model the absence of complex zeros analyticity of the free energy density

deterministic polynomial time approximation algorithms for partition functions and graph polynomials in this paper we show a new way of constructing deterministic polynomial time approximation algorithms for computing complex valued evaluations of a large class of graph polynomials on bounded degr

critical ising on the square lattice mixes in polynomial time the ising model is widely regarded as the most studied model of spin systems in statistical physics the focus of this paper is its dynamic stochastic version the glauber dynamics introduced in 1963 and by now the most popular means of sampling the ising measure intensive study throughout the last three decades has yielded a rigorous understanding of the spectral gap of the dynamics on mathbb z 2 everywhere except at criticality while the critical behavior of the ising model has long been the focus for physicists mathematicians have only recently developed an understanding of its critical geometry with the advent of sle cle and new tools to study conformally invariant systems

approach to equilibrium of glauber dynamics in the one phase region ii the general case we develop a new method based on renormalization group ideas block decimation procedure to prove under an assumption of strong mixing in a finite cube 0 a logarithmic sobolev inequality for the gibbs state of a discrete spin system as a consequence we derive the hypercontractivity of the markov semigroup of the associated glauber dynamics and the exponential convergence to equilibrium in the uniform norm in all volumes multiples of the cube 0

the glauber dynamics on colorings of a graph with high girth and maximum degree we prove that the glauber dynamics on the c colorings of a graph g on n vertices with girth g and maximum degree delta mixes rapidly if i c q delta and q q where q 1 4890 ldots is the root of 1 rm e 1 q 2 q rm e 1 q 1 and ii delta geq d log n and g geq d log delta for some constant d d q this improves the bound of roughly 1 763 delta obtained by dyer and frieze proceedings of the 32nd annual symposium on foundations of computer science 2001 for the same class of graphs our bound on this class of graphs is lower than the bound of 11 delta 6 approx1 833 delta obtained by vigoda j math phys 41 2000 pp 1555 1569 for general graphs

negative dependence and the geometry of polynomials we introduce the class of strongly rayleigh probability measures by means of geometric properties of their generating polynomials that amount to the stability of the latter this class covers important models such as determinantal measures e g product measures and uniform random spanning tree measures and distributions for symmetric exclusion processes we show that strongly rayleigh measures enjoy all virtues of negative dependence and we also prove a series of conjectures due to liggett pemantle and wagner respectively moreover we extend lyons recent results on determinantal measures and we construct counterexamples to several conjectures of pemantle and wagner on negative dependence and ultra log concave rank sequences

the complexity of approximately counting in 2 spin systems on k uniform bounded degree hypergraphs one of the most important recent developments in the complexity of approximate counting is the classification of the complexity of approximating the partition functions of antiferromagnetic 2 spin systems on bounded degree graphs this classification is based on a beautiful connection to the so called uniqueness phase transition from statistical physics on the infinite regular tree our objective is to study the impact of this classification on unweighted 2 spin models on k uniform hypergraphs as has already been indicated by yin and zhao the connection between the uniqueness phase transition and the complexity of approximate counting breaks down in the hypergraph setting nevertheless we show that for every non trivial symmetric k ary boolean function f there exists a degree bound 0 so that for all 0 the following problem is np hard given a k uniform hypergraph with maximum degree at most approximate the partition function of the hypergraph 2 spin model associated with f it is np hard to approximate this partition function even within an exponential factor by contrast if f is a trivial symmetric boolean function e g any function f that is excluded from our result then the partition function of the corresponding hypergraph 2 spin model can be computed exactly in polynomial time

multivariate pade approximants abstract for an operator f r n r analytic in the origin the notion of abstract multivariate pade approximant apa is introduced by making use of abstract polynomials the classical pade approximant n 1 is a special case of the multivariate theory and many interesting properties of classical pade approximants remain valid such as covariance properties and the block structure annie a m cuyt j oper theory6 2 1981 207 209 of the pade table also a projection property for multivariate pade approximants is proved

characterization of lee yang polynomials the lee yang circle theorem describes complex polynomials of degree n in z with all their zeros on the unit circle jzj d 1 these polynomials are obtained by taking z1 d d zn d z in certain multiaffine polynomials z1 zn which we call lee yang polynomials they do not vanish when jz1j jznj 1 we characterize the lee yang polynomials in nc1 variables in terms of polynomials in n variables those such that z1 zn 0 when jz1j jznj 1 including jzi j d 1 in a sense to be made precise later our current understanding of lee yang polynomials is based on the concept of asano contraction 1 we shall define an inner radius associated with a multiaffine polynomial and see that it behaves supermultiplicatively with respect to asano contraction proposition 2 using the properties of the inner radius we shall characterize the 2 lync1 nc 1 variables in terms of polynomials in n variables such that z1 zn 0 when jz1j jznj 1 theorem 3 this characterization will give us a good understanding of lync1 proposition 5 and allow us to exhibit elements of lync1 outside of the pair interaction class originally considered by lee and yang see in particular example 7 d the original

multivariate frobenius pade approximants properties and algorithms the aim of this paper is to construct rational approximants for multivariate functions given by their expansion in an orthogonal polynomial system this will be done by generalizing the concept of multivariate pade approximation after defining the multivariate frobenius pade approximants we will be interested in the two following problems the first one is to develop recursive algorithms for the computation of the value of a sequence of approximants at a given point the second one is to compute the coefficients of the numerator and denominator of the approximants by solving a linear system for some particular cases we will obtain a displacement rank structure for the matrix of the system we have to solve the case of a tchebyshev expansion is considered in more detail

helmholtz operators and symmetric space duality we consider the property of vanishing logarithmic term vlt for the fundamental solution of the shifted laplace d alembert operators b b a constant on pseudo riemannian reductive symmetric spaces m our main result is that such an operator on the c dual or flensted jensen dual of m has the vlt property if and only if a corresponding operator on m does for lorentzian spaces where the b are hyperbolic vlt is known to be equivalent to the strong huygens principle we use our results to construct a large supply of new space operator pairs satisfying huygens principle

on the distribution and gap structure of lee yang zeros for the ising model periodic and aperiodic couplings in this work we present some results on the distribution of lee yang zeros for the ferromagnetic ising model on the rooted cayley tree bethe lattice assuming free boundary conditions and in the one dimensional lattice with periodic boundary conditions in the case of the cayley tree we derive the conditions that the interactions between spins must obey in order to ensure existence or absence of phase transition at finite temperature t 0 the results are first obtained for periodic interactions along the generations of the lattice then using periodic approximants we are also able to obtain results for aperiodic sequences generated by substitution rules acting on a finite alphabet the particular examples of the fibonacci and the thue morse sequences are discussed most of the results are obtained for a cayley tree with arbitrary order d we will be concerned in showing whether or not the zeros become dense in the whole unit circle of the fugacity variable regarding the one dimensional ising model we derive a general treatment for the structure of gaps regions free of lee yang zeros around the unit circle

orthogonal polynomials on the unit circle with fibonacci verblunsky coefficients ii applications we consider cmv matrices with verblunsky coefficients determined in an appropriate way by the fibonacci sequence and present two applications of the spectral theory of such matrices to problems in mathematical physics in our first application we estimate the spreading rates of quantum walks on the line with time independent coins following the fibonacci sequence the estimates we obtain are explicit in terms of the parameters of the system in our second application we establish a connection between the classical nearest neighbor ising model on the one dimensional lattice in the complex magnetic field regime and cmv operators in particular given a sequence of nearest neighbor interaction couplings we construct a sequence of verblunsky coefficients such that the support of the lee yang zeros of the partition function for the ising model in the thermodynamic limit coincides with the essential spectrum of the cmv matrix with the constructed verblunsky coefficients under certain technical conditions we also show that the zeros distribution measure coincides with the density of states measure for the cmv matrix