delocalization of two dimensional random surfaces with hard core constraints we study the fluctuations of random surfaces on a two dimensional discrete torus the random surfaces we consider are defined via a nearest neighbor pair potential which we require to be twice continuously differentiable on a possibly infinite interval and infinity outside of this interval no convexity assumption is made and we include the case of the so called hammock potential when the random surface is uniformly chosen from the set of all surfaces satisfying a lipschitz constraint our main result is that these surfaces delocalize having fluctuations whose variance is at least of order log n where n is the side length of the torus we also show that the expected maximum of such surfaces is of order at least log n the main tool in our analysis is an adaptation to the lattice setting of an algorithm of richthammer who developed a variant of a mermin wagner type argument applicable to hard core constraints we rely also on the reflection positivity of the random surface model the result answers a question mentioned by brascamp et al on the hammock potential and a question of velenik

on the absence of breakdown of symmetry for the plane rotator model with long range random interaction we study the plane rotator model with hamiltonian frac 1 2 sum limits x ne y frac jxy cos mathcal o x mathcal o y left x y right 3 mathcal e where jxy for different pair x y are independent symmetric unbounded random variables it is proved that for almost all j all gibbs states p j are invariant by rotation

asymmetry in power law magnitude correlations time series of increments can be created in a number of different ways from a variety of physical phenomena for example in the phenomenon of volatility clustering well known in finance magnitudes of adjacent increments are correlated moreover in some time series magnitude correlations display asymmetry with respect to an increment s sign the magnitude of x i depends on the sign of the previous increment x i 1 here we define a model independent test to measure the statistical significance of any observed asymmetry we propose a simple stochastic process characterized by a an asymmetry parameter lambda and a method for estimating lambda we illustrate both the test and process by analyzing physiological data

fractionally integrated process with power law correlations in variables and magnitudes we propose a new stochastic process de ned by only two parameters 1 and 2 which generates power law correlations or anticorrelations in the variable increments and power law correlations in their magnitudes to account for empirical observations in a range of physical systems to demonstrate the utility of our process we consider the case with power law anticorrelations in the variable increments and power law correlations in their magnitudes to model scaling properties in heartbeat data the parameters 1 and 2 control separately the correlations in the increments and their magnitudes and we nd linear relations between each parameter and the corresponding correlation exponent

energy entropy inequalities for classical lattice systems a new characterization of equilibrium states for classical lattice systems is given in terms of correlation inequalities their physical meaning is found to express thermodynamic stability we demonstrate the applicability of the inequalities in specific models

strict convexity of the free energy for a class of non convex gradient models we consider a gradient interface model on the lattice with interaction potential which is a non convex perturbation of a convex potential we show using a one step multiple scale analysis the strict convexity of the surface tension at high temperature this is an extension of funaki and spohn s result 8 where the strict convexity of potential was crucial in their proof

real vs complex rational chebyshev approximation on an interval if e c l 1 is real valued let er f and be the errors in best approximation to in the supremum norm by rational functions of type but for no n 1 is it known whether ym inff e f er f is zero or strictly positive here we show that both are possible y0i 0 but y 0 for n m 3 related results are obtained for approximation on regions in the plane

on equivalence of spin and field pictures of lattice systems we investigate the spin and field systems on a lattice connected by the kac siegert transform it is shown that the structures of corresponding theories are equivalent in the sense of isomorphy of space of gibbs states and order parameters using the idea of equivalence of spin and field pictures we exhibit a class of lattice systems possessing infinitely uncountably many ground states the systems of this type with infinite range slow decaying interactions are expected to have a spin glass phase transition

statistical properties of the volatility of price fluctuations we study the statistical properties of volatility measured by locally averaging over a time window t the absolute value of price changes over a short time interval ensuremath delta t we analyze the s 500 stock index for the 13 year period jan 1984 to dec 1996 we find that the cumulative distribution of the volatility is consistent with a power law asymptotic behavior characterized by an exponent ensuremath mu ensuremath approx 3 similar to what is found for the distribution of price changes the volatility distribution retains the same functional form for a range of values of t further we study the volatility correlations by using the power spectrum analysis both methods support a power law decay of the correlation function and give consistent estimates of the relevant scaling exponents also both methods show the presence of a crossover at approximately 1 5 days in addition we extend these results to the volatility of individual companies by analyzing a data base comprising all trades for the largest 500 u s companies over the two year period jan 1994 to dec 1995

weighted h 2 rational approximation and consistency we investigate consistency properties of rational approximation of prescribed type in the weighted hardy space h 2 mu for the exterior of the unit disk where mu is a positive symmetric measure on the unit circle mathbb t the question of consistency which is especially significant for gradient algorithms that compute local minima concerns the uniqueness of critical points in the approximation criterion for the case when the approximated function is itself rational in addition to describing some basic properties of the approximation problem we prove for measures mu having a rational function distribution weight with respect to arclength on mathbb t that consistency holds only under rather restricted conditions

volatility distribution in the s p500 stock index abstract we study the volatility of the s p500 stock index from 1984 to 1996 and find that the volatility distribution can be very well described by a log normal function further using detrended fluctuation analysis we show that the volatility is power law correlated with hurst exponent 0 9

varieties of long memory models long memory is defined as a series having a slowly declining correlogram or equivalently an infinite spectrum at zero frequency fractional integrated processes have such properties but here it is pointed out that a number of other processes can also be long memory including generalized fractionally integrated models arising from aggregation time changing coefficient models and possibly nonlinear models it seems that there are many classes of processes that deserve further study the relevance of long memory is illustrated using absolute returns from a daily stock market index

a new proof of ruelle s superstability bounds for a slightly stronger assumptions on the interaction we give a very transparent proof of ruelle s result in the language of poisson integral measure representation for the correlation functions on the configuration space using some kind of cluster expansion in the densities of configurations

on the absence of spontaneous symmetry breaking and of crystalline ordering in two dimensional systems we develop a unified approach based on araki s relative entropy concept to proving absence of spontaneous breaking of continuous internal symmetries and translation invariance in two dimensional statistical mechanical systems more precisely we show that under rather general assumptions on the interactions all equilibrium states of a two dimensional system have all the symmetries compact internal and spatial of the dynamics except possibly rotation invariance rotation invariance remains unbroken if connected correlations decay more rapidly than the inverse square distance we also prove that two dimensional systems with a non compact internal symmetry group like anharmonic crystals typically do not have gibbs states

energy correlations in o n models and the wolff representation we prove that energy functions are positively correlated in isotropic ferromagnetic o n models on an arbitrary graph in our inductive proof this is used to prove the strong fkg property of the wolff representation for isotropic ferromagnetic o n 1 models this strong fkg property is then used to prove energy correlations for the o n 1 model furthermore percolation in the wolff representation is proved to be a necessary and sufficient condition for positivity of the spontaneous magnetization previously known only for n 3

real and complex chebyshev approximation on the unit disk and interval we announce the resolution of a number of outstanding questions regarding real and complex chebyshev supremum norm approximation by rational functions on a disk and on an interval the proofs consist mainly of symmetry arguments applied to explicit examples the most important results complex rational best approximations on a disk are in general not unique real functions on an interval can in general be approximated arbitrarily much better by complex rational functions than by real ones details will appear in 3 8

critical behavior in a quasi d dimensional spin model we study a classical spin model more precisely a class of models with o n symmetry that can be viewed as a simplified d dimensional lattice model it is equivalent to a non translationinvariant one dimensional model and contains the dimensionality d as a parameter that need not be an integer the critical dimension turns out to be 2 just as in the usual translation invariant models we study the phase structure critical phenomena and spontaneous symmetry breaking furthermore we compute the perturbation expansion to low order with various boundary conditions in our simplified models a number of questions can be answered that remain controversial in the translation invariant models such as the asymptoticity of the perturbation expansion and the role of super instantons we find that perturbation theory produces the right asymptotic expansion in dimension d 2 only with special boundary conditions finally the model allows a test of the percolation ideas of patrascioiu and seiler

u 1 su 2 gauge invariance of non relativistic quantum mechanics and generalized hall effects we show that the non relativistic quantum mechanics of particles with spin coupled to an electromagnetic field has a naturalu 1 su 2 gauge invariance ward identities reflecting this gauge invariance combined with an assumption of incompressibility of a system of such particles in an appropriate external field and for suitable values of the particle density permit us to determine the form of the effective action of the system as a functional of small fluctuations in the electromagnetic field in the large distance adiabatic limit in this limit the action is found to have a universal form we present explicit results for two dimensional incompressible electron fluids and apply them to derive the equations of linear response theory describing a variety of generalized hall effects sum rules for the hall conductivities magnetic susceptibilities and other quantities of physical interest are found

on the symmetry of the gibbs states in two dimensional lattice systems under fairly general conditions if a two dimensional classical lattice system has an internal symmetry groupg which is a compact connected lie group then all gibbs states areg invariant

the mermin wagner phenomenon and cluster properties of one and two dimensional systems we give optimal conditions concerning the range of interactions for the absence of spontaneous breakdown of continuous symmetries for one and two dimensional quantum and classical lattice and continuum systems for a class of models verifying infrared bounds our conditions are necessary and sufficient using the same techniques we obtain a priori bounds on clustering for systems with continuous symmetry improving results of jasnow and fisher

complex approximation of real functions by reciprocals of polynomials characterizations are given for local and global best rational approximations to a real function the characterizations are specialized to reciprocals of polynomials where they are used to settle some conjectures and questions introduction let f be a continuous real valued function defined on a real interval let r be a real valued rational function in rm if r is a local best approximation to f from rm it is a global best approximation if the same r is also a local n best approximation to f from the complex rational functions rm c it is not known if r must be a global best approximation from rmn c to state known results let crit f r x l f r x l ilf rll rcrit p e pm q e pn and x q x o n crit f r 0 and fc cf 1 c r cer an oscillation of length k is an increasing subset xj k of crit f r for which f r xj f r xj with these definitions we outline the framework relevant to this paper from the classical theory r e is a best q approximation to f from rm if and only if there is an oscillation of length n max m degq n degp 2 cheney 1966 if there is an oscillation of length m n 2 then for all sufficiently small c 0 r is a best approximation to fc from rm c wulbert 1980 if there is an oscillation of length m 2n 2 then r is a best approximation to f from 9irm c and consequently is the unique best approximation to fc for all c wulbert 1978 a local best approximation to fc for all positive c is a best approximation to f from 9irm c wulbert 1988 a local best approximation to f from rm c is a best approximation on crit f r to f from rcrit this fact was known to a ruttan ruttan 1985 we will also include a short proof below lemma 4 suppose card crit f r m n 2 if r is a best approximation on crit f r to f from rcrit then r is a global best approximation to f from received by the editors may 11 1988 1980 mathematics subject classification 1985 revision primary 41a27 1989 american mathematical society 0002 9947 89 1 00 25 per page

antiferromagnetic potts models on the square lattice a high precision monte carlo study we study the antiferromagnetic q state potts model on the square lattice for q 3 and q 4 using the wang swendsen koteck wsk monte carlo algorithm and a powerful finite size scaling extrapolation method for q 3 we obtain good control up to correlation length 5000 the data are consistent with ae2 p 1 a1e as with p 1 the staggered susceptibility behaves as stagg 5 3 for q 4 the model is disordered 2 even at zero temperature in appendices we prove a correlation inequality for potts antiferromagnets on a bipartite lattice and we prove ergodicity of the wsk algorithm at zero temperature for potts antiferromagnets on a bipartite lattice

discontinuity of the spin wave stiffness in the two dimensional xy model using a graphical representation based on the wolff algorithm the classical d dimensional xy model and some related spin systems are studied it is proved that in d 2 the predicted discontinuity in the spin wave stiffness indeed occurs further the critical properties of the spin system are related to percolation properties of the graphical representation in particular a suitably defined notion of percolation in the graphical representation is proved to be the necessary and sufficient condition for positivity of the spontaneous magnetization

quantum energy entropy inequalities a new method for proving the absence of symmetry breaking for quantum systems we develop a new method based on a general energy entropy inequality to rule out spontaneous breaking of symmetries the main advantage of our scheme consists in its clear cut physical significance and its new areas of applicability in particular we can handle discrete symmetry groups as well as continuous ones finally a few illustrations are discussed

regularity properties and pathologies of position space renormalization group transformations scope and limitations of gibbsian theory we reconsider the conceptual foundations of the renormalization group rg formalism and prove some rigorous theorems on the regularity properties and possible pathologies of the rg map our main results apply to local in position space rg maps acting on systems of bounded spins compact single spin space regarding regularity we show that the rg map defined on a suitable space of interactions formal hamiltonians is always single valued and lipschitz continuous on its domain of definition this rules out a recently proposed scenario for the rg description of first order phase transitions on the pathological side we make rigorous some arguments of griffiths pearce and israel and prove in several cases that the renormalized measure is not a gibbs measure for any reasonable interaction this means that the rg map is ill defined and that the conventional rg description of first order phase transitions is not universally valid for decimation or kadanoff transformations applied to the ising model in dimension d 3 these pathologies occur in a full neighborhood 0 h e of the low temperature part of the first order phase transition surface for block averaging transformations applied to the ising model in dimension d 2 the pathologies occur at low temperatures for arbitrary magnetic field strength pathologies may also occur in the critical region for ising models in dimension d 4 we discuss the heuristic and numerical evidence on rg pathologies in the light of our rigorous theorems in addition we discuss critically the concept of gibbs measure which is at the heart of present day classical statistical mechanics we provide a careful and we hope pedagogical overview of the theory of gibbsian measures as well as the less familiar non gibbsian measures emphasizing the distinction between these two objects and the possible occurrence of the latter in different physical situations we give a rather complete catalogue of the known examples of such occurrences the main message of this paper is that despite a well established tradition gibbsiannessshould not be taken for granted

the isotropic o 3 model and the wolff representation for the isotropic o 3 model we prove that percolation in the wolff representation is a necessary and sufficient condition for positivity of the spontaneous magnetization

graphical representations and cluster algorithms ii we continue the study initiated in part i of graphical representations and cluster algorithms for various models in or related to statistical mechanics for certain models e g the blume emery griffths model and various generalizations we develop fortuin kasteleyn type representations which lead immediately to swendsen wang type algorithms for other models e g the random cluster model that are defined by a graphical representation we develop cluster algorithms without reference to an underlying spin system in all cases phase transitions are related to percolation or incipient percolation in the graphical representation which via the ic algorithm allows for the rapid simulation of these systems at the transition point pertinent examples include the continuum widom rowlinson model the restricted 1 step solid on solid model and the xy model

phase structure of two dimensional spin models and percolation for a class of classical spin models in 2d satisfying a certain continuity constraint it is proven that some of their correlations do not decay exponentially the class contains discrete and continuous spin systems with abelian and non abelian symmetry groups for the discrete models our results imply that they show either long range order or are in a soft phase characterized by powerlike decay of correlations for the continuous models only the second possibility exists the continuous models include a version of the plane rotator o 2 model for this model we rederive modulo two conjectures the frohlich spencer result on the existence of the kosterlitz thouless phase in a very simple way the proof is based on percolation theoretic and topological arguments

on bandlimitedness and lag limitedness of fractional gaussian noise abstract the contributions of this short paper are two fold we shall show two interesting properties of fractional gaussian noise fgn namely its bandlimitedness and lag limitedness the computation formulas for the maximum frequency of bandlimited fgn and the maximum lag of lag limited fgn are proposed in addition we will give a new explanation of the statistical dependences of fgn based on the present bandlimitedness and lag limitedness of fgn

robust phase transitions for heisenberg and other models on general trees we study several statistical mechanical models on a general tree particular attention is devoted to the classical heisenberg models where the state space is the d dimensional unit sphere and the interactions are proportional to the cosines of the angles between neighboring spins the phenomenon of interest here is the classification of phase transition non uniqueness of the gibbs state according to whether it is robust in many cases including all of the heisenberg and potts models occurrence of robust phase transition is determined by the geometry branching number of the tree in a way that parallels the situation with independent percolation and usual phase transition for the ising model the critical values for robust phase transition for the heisenberg and potts models are also calculated exactly in some cases such as the q 3 potts model robust phase transition and usual phase transition do not coincide while in other cases such as the heisenberg models we conjecture that robust phase transition and usual phase transition are equivalent in addition we show that symmetry breaking is equivalent to the existence of a phase transition a fact believed but not known for the rotor model on z 2