random infinite volume gibbs states for the curie weiss random field ising model an approach to the definition of infinite volume gibbs states for the quenched random field ising model is considered in the case of a curie weiss ferromagnet it turns out that these states are random quasi free measures they are random convex linear combinations of the free product measures shifted by the corresponding effective mean fields the conditional self averaging property of the magnetization related to this randomness is also discussed

analogues of non gibbsianness in joint measures of disordered mean field models it is known that the joint measures on the product of spin space and disorder space are very often non gibbsian measures for lattice systems with quenched disorder at low temperature are there reflections of this non gibbsianness in the corresponding mean field models we study the continuity properties of the conditional probabilities in finite volume of the following mean field models a joint measures of random field ising b joint measures of dilute ising c decimation of ferromagnetic ising the conditional probabilities are functions of the empirical mean of the conditionings so we look at the large volume behavior of these functions to discover non trivial limiting objects for a we find 1 discontinuous dependence for almost any realization and 2 dependence of the conditional probabilities on the phase in contrast to that we see continuous behavior for b and c for almost any realization this is in complete analogy to the behavior of the corresponding lattice models in high dimensions it shows that non gibbsian behavior which seems a genuine lattice phenomenon can be partially understood already on the level of mean field models

cluster expansion for abstract polymer models a new direct proof of convergence of cluster expansions for polymer contour models is given in an abstract setting it does not rely on kirkwood salsburg type equations or combinatorics of trees a distinctive feature is that at all steps the considered clusters contain every polymer at most once

on the gibbsian nature of the random field kac model under block averaging we consider the kac ising model in an arbitrary configuration of local magnetic fields left eta kern 1pt i right i quad quad in quad kern 1pt quad mathbb z d in any dimension d at any inverse temperature we investigate the gibbs properties of the renormalized infinite volume measures obtained by block averaging any of the gibbs measures corresponding to fixed with block length small enough compared to the range of the kac interaction we show that these measures are gibbs measures for the same renormalized interaction potential this potential depends locally on the field configuration and decays exponentially uniformly in for which we give explicit bounds the construction of the potential is based on a high temperature type cluster expansion

infinite dimensional diffusion processes as gibbs measures on c 0 1 z d an infinite lattice system of interacting diffusion processes is characterized as a gibbs distribution on c 0 1 z d with continuous local conditional probabilities using estimates for the vasserstein metric onc 0 1 dobrushin s contraction technique is applied in order to obtain information about macroscopic properties of the entire diffusion process

improved perturbation expansion for disordered systems beating griffiths singularities we introduce a new expansion to prove exponential clustering of connected correlations in a large class of disordered systems our expansion converges for values of the temperature and magnetic field where standard cluster expansions diverge due to the presence of griffiths type singularities it is organized inductively over an infinite sequence of increasing distance scales in each induction step one redefines what is means by the unperturbed system a procedure somewhat reminiscent of k a m theory our techniques may be useful in dealing with the so called large field problem in real space renormalization group schemes

typical configurations for one dimensional random field kac model in this paper we study the typical profiles of a random field kac model we give upper and lower bounds of the space scale where the profiles are constant the results hold almost surely with respect to the realizations of the random field the analysis is based on a block spin construction deviation techniques for the local empirical order parameters and concentration inequalities for the realizations of the random magnetic field for the upper bound we exhibit a scale related to the law of the iterated logarithm where the random field makes an almost sure fluctuation that obliges the system to break its rigidity for the lower bound we prove that on a smaller scale the fluctuations are not strong enough to allow this transition

infinite differentiability for one dimensional spin system with long range random interaction we consider one dimensional spin systems with hamiltonian h left sigma lambda right sum limits t t in lambda frac varepsilon tt left t t right alpha sigma t sigma t h sum limits t in lambda sigma t where tt are independent random variables and using decimation and the cluster expansion we show that when 3 2 ande tt 0 for any magnetic fieldh and inverse temperature the correlation functions and the free energy arec both inh and

renormalization group in the uniqueness region weak gibbsianity and convergence we analyze the block averaging transformation applied to lattice gas models with short range interaction in the uniqueness region below the critical temperature we prove weak gibbsianity of the renormalized measure and convergence of the renormalized potential in a weak sense since we are arbitrarily close to the coexistence region we have a diverging characteristic length of the system the correlation length or the critical length for metastability or both thus to perturbatively treat the problem we have to use a scale adapted expansion moreover such a model below the critical temperature resembles a disordered system in the presence of griffiths singularity then the cluster expansion that we use must be graded with its minimal scale length diverging when the coexistence line is approached

the order parameter in a spin glass various possible precise definitions of an edwards anderson type of order parameter for an ising model spin glass are considered using boundary conditions for a finite system states of an infinite system and a duplicate system approach several of these definitions are shown to yield identical results

relative entropy and variational properties of generalized gibbsian measures we study the relative entropy density for generalized gibbs measures we first show its existence and obtain a familiar expression in terms of entropy and relative energy for a class of almost gibbsian measures almost sure continuity of conditional probabilities for quasilocal measures we obtain a full variational principle for the joint measures of the random field ising model we show that the weak gibbs property holds with an almost surely rapidly decaying translation invariant potential for these measures we show that the variational principle fails as soon as the measures lose the almost gibbs property these examples suggest that the class of weakly gibbsian measures is too broad from the perspective of a reasonable thermodynamic formalism

corrections to the critical temperature in 2d ising systems with kac potentials we consider ad 2 ising system with a kac potential whose mean field critical temperature is 1 calling 0 the kac parameter we prove that there existsc 0 so that the true inverse critical temperature cr 1 by2 log 1 for anyb c and correspondingly small we also show that if 0 andb c suitably then the correlation functions normalized and rescaled converge to those of a non gaussian euclidean field theory

thermodynamics and statistical analysis of gaussian random fields gaussian fields are considered as gibbsian fields thermodynamic functions are calculated for them and the variational principle is proved as an application we get an approximation of log likelihood and an information theoretic interpretation of the asymptotic behaviour of the maximum likelihood estimator for gaussian markov fields

multidimensional random subshifts of finite type and their large deviations i introduce random multidimensional subshifts of finite type which generalize models of spin glasses and establish the almost sure large deviations bounds for gibbs measures there the paper is sequel to ekw where the corresponding results were obtained for deterministic multidimensional subshifts of finite type

weighted association rule mining via a graph based connectivity model association rule mining is an important data mining task that discovers relationships among items in a transaction database classical association rule mining approaches make the implicit assumption that an item s importance is determined by its support in contrast weighted association rule mining warm attempts to provide a notion of importance or weight to individual items that are not based solely on item support previous approaches to weighted association rule mining assign item weights in a subjective manner based on a user s specialized knowledge of the underlying domain that is involved such approaches are infeasible when millions of items are present in a dataset or when domain knowledge is unavailable furthermore even when such domain information is available a weight assignment based on subjective information constrains the knowledge discovered to fit with the weights assigned thus inhibiting the discovery of new trends in the data in this research we automate the process of weight assignment by formulating a linear model that captures relationships between items this approach extends prior research based on the valency model we extend the valency model by expanding the field of interaction beyond immediate neighborhoods and show that this leads to significant improvements in performance on a number of different metrics that we use

on the mean field ising model in a random external field we use a method developed by van hemmen to obtain the free energy of the mean field ising model in a random external magnetic field some results of previous mean field calculations are confirmed and generalized the tricritical point in the global phase diagram is discussed in detail we also consider different probability distributions of the random fields and provide some proofs regarding the conditions for the existence of a tricritical point

large deviations for the empirical field of a gibbs measure on applique le principe de contraction pour obtenir un principe de grande deviation pour la distribution des distributions empiriques on donne une description detaillee de la fonction de taux resultante

derivation of hartree s theory for generic mean field bose systems in this paper we provide a novel strategy to prove the validity of hartree s theory for the ground state energy of bosonic quantum systems in the mean field regime for the well known case of trapped bose gases this can be shown using the strong quantum de finetti theorem which gives the structure of infinite hierarchies of k particles density matrices here we deal with the case where some particles are allowed to escape to infinity leading to a lack of compactness our approach is based on two ingredients 1 a weak version of the quantum de finetti theorem and 2 geometric techniques for many body systems our strategy does not rely on any special property of the interaction between the particles in particular our results cover those of benguria lieb and lieb yau for respectively bosonic atoms and boson stars

gibbs non gibbs transitions via large deviations computable examples we give new and explicitly computable examples of gibbs non gibbs transitions of mean field type using the large deviation approach introduced in van enter et al in mosc math j 10 687 711 2010 these examples include brownian motion with small variance and related diffusion processes such as the ornstein uhlenbeck process as well as birth and death processes we show for a large class of initial measures and diffusive dynamics both short time conservation of gibbsianness and dynamical gibbs non gibbs transitions

pair wise association measures for categorical and mixed data we introduce two measures for the strength of the association between two categorical variables the measures denoted by 1 and 2 take values in the interval 0 1 a value of zero means there is no association between the two categorical variables while a value of 1 means there is a perfect association e g when we associate a variable with itself we obtain 1 the measures are symmetric with respect to the order of variables invariant with respect to permutations of the categories of the variables and scalable for large number of observations in addition extensions of the proposed measures are presented for measuring the strength of association between pair of mixed variables one quantitative and the other is categorical the performance of the proposed measures compared to other association measures is investigated using simulated as well as real data

central limit theorem for an infinite lattice system of interacting diffusion processes on donne une demonstration basee sur une representation par integrale stochastique de dimension infinie des fonctionnelles lisses des processus de diffusion

smoothing low snr molecular images via anisotropic median diffusion we propose a new anisotropic diffusion filter for denoising low signal to noise molecular images this filter which incorporates a median filter into the diffusion steps is called an anisotropic median diffusion filter this hybrid filter achieved much better noise suppression with minimum edge blurring compared with the original anisotropic diffusion filter when it was tested on an image created based on a molecular image model the universal quality index proposed in this paper to measure the effectiveness of denoising suggests that the anisotropic median diffusion filter can retain adherence to the original image intensities and contrasts better than other filters in addition the performance of the filter is less sensitive to the selection of the image gradient threshold during diffusion thus making automatic image denoising easier than with the original anisotropic diffusion filter the anisotropic median diffusion filter also achieved good denoising results on a piecewise smooth natural image and real raman molecular images

segmentation of natural images using anisotropic diffusion and linking of boundary edges we present a segmentation method of natural images that uses an anisotropic diffusion algorithm and a region growing algorithm we propose a modified version of the anisotropic diffusion algorithm as a precise edge preserving smoothing technique modified by using boundary edges we incorporate a linking algorithm for boundary edges based on a directional potential function into the anisotropic diffusion algorithm to improve the ability of edge preserving smoothing as a result unnecessary details of images are effectively smoothed before performing a region growing algorithm therefore the proposed method is suitable for an accurate segmentation of natural images several simulated examples are presented that demonstrate the effectiveness of the proposed technique

on secondary bifurcations for some nonlinear convolution equations etude de quelques equations de convolution non lineaires sur le tore a d dimensions conditions d existence de bifurcations secondaires

low temperature dynamics of the curie weiss model periodic orbits multiple histories and loss of gibbsianness we consider the curie weiss model at initial temperature 0 1 in vanishing external field evolving under a glauber spin flip dynamics with temperature 0 1 we study the limiting conditional probabilities and their continuity properties and discuss their set of points of discontinuity bad points we provide a complete analysis of the transition between gibbsian and non gibbsian behavior as a function of time extending earlier work for the case of independent spin flip dynamics

on the free energy of the hopfield model the general theory of inhomogeneous mean field systems of raggio and werner provides a variational expression for the almost sure limiting free energy density of the hopfield model h n p xi s frac 1 2n sum limits i j 1 n sum limits mu 1 n xi i mu xi j mu s i s j for ising spinssi andp random patterns 1 2 n under the assumption that mathop lim limits n to gamma n 1 sum limits i 1 n delta xi i lambda xi i xi i 1 xi i 2 xi i p exists almost surely in the space of probability measures overp copies of 1 1 including an external field term p h i 1 n i si we give a number of general properties of the free energy density and compute it for a p 2 in general and b p arbitrary when is uniform and at most the two componentsh 1 andh 2 are nonzero obtaining the almost sure formula f beta h tfrac 1 2 f ew beta h mu 1 h mu 2 tfrac 1 2 f ew beta h mu 1 h mu 2 for the free energy wherefcw denotes the limiting free energy density of the curie weiss model with unit interaction constant in both cases we obtain explicit formulas for the limiting almost sure values of the so called overlap parameters m n mu beta h n 1 sum limits i 1 n xi i mu left langle s i right rangle in terms of the curie weiss magnetizations for the general i i d case with prob i 1 1 2 we obtain the lower bound 1 4 2 p 1 for the temperaturetc separating the trivial free regime where the overlap vector is zero from the nontrivial regime where it is nonzero this lower bound is exact forp 2 or e 0 or e 1 2 forp 2 we identify an intermediate temperature region between t 1 4 2 and tc 1 4 2 where the overlap vector is homogeneous i e all its components are equal and nonzero t marks the transition to the nonhomogeneous regime where the components of the overlap vector are distinct we conjecture that the homogeneous nonzero regime exists forp 3 and that t max 1 4 2 p 1 0

short time gibbsianness for infinite dimensional diffusions with space time interaction we consider a class of infinite dimensional diffusions where the interaction between the components has a finite extent both in space and time we start the system from a gibbs measure with a finite range uniformly bounded interaction under suitable conditions on the drift we prove that there exists t0 0 such that the distribution at time t t0 is a gibbs measure with absolutely summable interaction the main tool is a cluster expansion of both the initial interaction and certain time reversed girsanov factors coming from the dynamics

a perturbed mean field model of an interacting boson gas and the large deviation principle this is a study of the equilibrium thermodynamics of a mean field model of an interacting boson gas perturbed by a term quadratic in the occupation numbers of the free gas energy levels we prove the existence of the pressure in the thermodynamic limit we obtain also a variational formula for the pressure this enables us to compare the effect of a smooth quadratic perturbation with that of a singular one the huang yang luttinger model the proof uses a large deviation result for the occupation measure of the free boson gas which is of independent interest

residuals and goodness of fit tests for stationary marked gibbs point processes the inspection of residuals is a fundamental step to investigate the quality of adjustment of a parametric model to data for spatial point processes the concept of residuals has been recently proposed by baddeley et al 2005 as an empirical counterpart of the it campbell equilibrium equation for marked gibbs point processes the present paper focuses on stationary marked gibbs point processes and deals with asymptotic properties of residuals for such processes in particular the consistency and the asymptotic normality are obtained for a wide class of residuals including the classical ones raw residuals inverse residuals pearson residuals based on these asymptotic results we define goodness of fit tests with type i error theoretically controlled one of these tests constitutes an extension of the quadrat counting test widely used to test the null hypothesis of a homogeneous poisson point process

a simple inductive approach to the problem of convergence of cluster expansions of polymer models we explain a simple inductive method for the analysis of the convergence of cluster expansions taylor expansions mayer expansions for the partition functions of polymer models we give a very simple proof of the dobrushin koteck preiss criterion and formulate a generalization usable for situations where a successive expansion of the partition function has to be used

renormalization group transformations near the critical point some rigorous results we consider renormalization group rg transformations for classical ising type lattice spin systems in the infinite volume limit formally the rg maps a hamiltonian h into a renormalized hamiltonian h exp h t exp h where t denotes a specific rg probability kernel t 1 for every configuration with the help of the dobrushin uniqueness condition and standard results on the polymer expansion haller and kennedy gave a sufficient condition for the existence of the renormalized hamiltonian in a neighborhood of the critical point by a more complicated but reasonably straightforward application of the cluster expansion machinery the present investigation shows that their condition would further imply a band structure on the matrix of partial derivatives of the renormalized interaction with respect to the original interaction this in turn gives an upper bound for the rg linearization

some remarks on ultracontractivity we study the semi group t t generated by the operator 12 u on the lebesgue space l2 rd with the measure e u we prove via different methods using probabilistic techniques or pde arguments that t t is ultracontractive i e for t 0 it maps l1 into l when the function u satisfies a growth condition at infinity which is essentially for instance when the dimension d 1 the integrability of 1 u at infinity also we consider the analogous properties of the semi group generated by the fractional powers of the above operator

the yang yang thermodynamic formalism and large deviations the partition function for a one dimensional system of bosons with repulsive delta function interaction is investigated we prove that if the bethe ansatz eigenfunctions form a complete set then the grand canonical pressure is given by the yang yang formula the proof uses a probabilistic formalism to express the partition function as an expectation with respect to a probability measure on a banach space of measures the asymptotic behaviour of the expectation in the thermodynamic limit is determined by the large deviation principle this method is applicable in situations in which the hamiltonian can be diagonalised using the bethe ansatz

propagation of gibbsianness for infinite dimensional gradient brownian diffusions we study the strong gibbsian character on mathbb r mathbb z d of the law at time t of an infinite imensional gradient brownian diffusion when the initial distribution is gibbsian

large deviations for gibbs random fields a large deviation principle for gibbs random fields on zd is proven and a corresponding large deviations proof of the gibbs variational formula is given a generalization of the lanford theory of large deviations is also obtained

metastates in disordered mean field models ii the superstates we continue to investigate the size dependence of disordered mean field models with finite local spin space in more detail illustrating the concept of superstates as recently proposed by bovier and gayrard we discuss various notions of convergence for the behavior of the paths t tn t 0 1 in the thermodynamic limit n here n is the gibbs measure in the finite volume 1 n and is the disorder variable in particular we prove refined convergence statements in our concrete examples the hopfield model with finitely many patterns having continuous paths and the curie weiss random field ising model having singular paths

weakly gibbsian representations for joint measures of quenched lattice spin models can the joint measures of quenched disordered lattice spin models with finite range on the product of spin space and disorder space be represented as suitably generalized gibbs measures of an annealed system we prove that there is always a potential depending on both spin and disorder variables that converges absolutely on a set of full measure w r t the joint measure weak gibbsianness this positive result is surprising when contrasted with the results of a previous paper k6 where we investigated the measure of the set of discontinuity points of the conditional expectations investigation of a s gibbsianness in particular we gave natural negative examples where this set is even of measure one including the random field ising model further we discuss conditions giving the convergence of vacuum potentials and conditions for the decay of the joint potential in terms of the decay of the disorder average over certain quenched correlations we apply them to various examples from this one typically expects the existence of a potential that decays superpolynomially outside a set of measure zero our proof uses a martingale argument that allows to cut an infinite volume analogue of the quenched free energy into local pieces along with generalizations of kozlov s constructions

hypercontractivity and spectral gap of symmetric diffusions with applications to the stochastic ising models abstract we describe some results relating the spectral gap and logarithmic sobolev constants we work initially in the context of symmetric diffusions on a finite dimensional manifold and later apply our results to the analysis of certain infinite systems of mildly interacting diffusions

orthogonality and completeness of the bethe ansatz eigenstates of the nonlinear schroedinger model a rigorous proof is given of the orthogonality and the completeness of the bethe ansatz eigenstates of then body hamiltonian of the nonlinear schroedinger model on a finite interval the completeness proof is based on ideas of c n yang and c p yang but their continuity argument at infinite coupling is replaced by operator monotonicity at zero coupling the orthogonality proof uses the algebraic bethe ansatz method or inverse scattering method applied to a lattice approximation introduced by izergin and korepin the latter model is defined in terms of monodromy matrices without writing down an explicit hamiltonian it is shown that the eigenfunctions of the transfer matrices for this model converge to the bethe ansatz eigenstates of the nonlinear schroedinger model

phase transition in the 3d random field ising model we show that the three dimensional ising model coupled to a small random magnetic field is ordered at low temperatures this means that the lower critical dimension dl for the theory isdl 2 settling a long controversy on the subject our proof is based on an exact renormalization group rg analysis of the system this analysis is carried out in the domain wall representation of the system and it is inspired by the scaling arguments of imry and ma the rg acts in the space of ising models and in the space of random field distributions driving the former to zero temperature and the latter to zero variance

metastates in disordered mean field models random field and hopfield models we rigorously investigate the size dependence of disordered mean field models with finite local spin space in terms of metastates thereby we provide an illustration of the framework of metastates for systems of randomly competing gibbs measures in particular we consider the thermodynamic limit of the empirical metastate 1 n sum nolimits n 1 n delta mu eta eta where n is the gibbs measure in the finite volume 1 n and the frozen disorder variable is fixed we treat explicitly the hopfield model with finitely many patterns and the curie weiss random field ising model in both examples in the phase transition regime the empirical metastate is dispersed for largen moreover it does not converge for a e but rather in distribution for whose limits we given explicit expressions we also discuss another notion of metastates due to aizenman and wehr

une approche gibbsienne des diffusions browniennes infini dimensionnelles nous montrons qu il y a correspondance biunivoque entre les lois de diffusions browniennes infini dimensionnelles et les etats de gibbs sur l espace des trajectoires omega c 0 1 mathbb z d ce resultat est ensuite applique aux problemes de transition de phases du retournement du temps et a l etude des mesures reversibles le principal outil est une caracterisation des etats de gibbs par une version infini dimensionnelle de la formule d integration par parties du calcul de malliavin

global specifications and nonquasilocality of projections of gibbs measures reference gr pf article 1997 001doi 10 1214 aop 1024404514view record in web of science record created on 2008 11 27 modified on 2017 05 12

graded cluster expansion for lattice systems in this paper we develop a general theory which provides a unified treatment of two apparently different problems the weak gibbs property of measures arising from the application of renormalization group maps and the mixing properties of disordered lattice systems in the griffiths phase we suppose that the system satisfies a mixing condition in a subset of the lattice whose complement is sparse enough namely large regions are widely separated we then show how it is possible to construct a convergent multi scale cluster expansion

fluctuation results for general block spin ising models we study a block spin mean field ising model i e a model of spins in which the vertices are divided into a finite number of blocks with each block having a fixed proportion of vertices and where pair interactions are given according to their blocks for the vector of block magnetizations we prove large deviation principles and central limit theorems under general assumptions for the block interaction matrix using the exchangeable pair approach of stein s method we establish a rate of convergence in the central limit theorem for the block magnetization vector in the high temperature regime

variational principle and almost quasilocality for renormalized measures we study the variational principle for some non gibbsian measures we give a necessary and sufficient condition for the validity of the implication zero relative entropy density implies common version of conditional probabilities so called second part of the variational principle applying this to noisy decimations of the low temperature phases of the ising model we obtain almost sure quasilocality for these measures and the second part of the variational principle for the projection of low temperature ising phases on a one dimensional layer we also obtain the second part of the variational principle

robustness of the non gibbsian property some examples we discuss some examples of measures on lattice systems which lack the property of being a gibbs measure in a rather strong sense

loss and recovery of gibbsianness for xy models in external fields we consider planar rotors xy spins in zd starting from an initial gibbs measure and evolving with infinite temperature stochastic diffusive dynamics at intermediate times if the system starts at low temperature gibbsianness can be lost due to the influence of the external initial field gibbsianness can be recovered after large finite times we prove some results supporting this picture

projections of gibbs measures may be non gibbsian consider the phase of the two dimensional nearest neighbor ferromagnetic ising model at a temperature belowt c let be the restriction of this measure to a coordinate axis we prove that there is no one dimensional translation invariant summable interaction for which is a gibbs measure this is proven by showing that if such an interaction existed would have large deviation properties different from those it actually has percolation methods are used in the proof

loss without recovery of gibbsianness during diffusion of continuous spins we consider a specific continuous spin gibbs distribution t 0 for a double well potential that allows for ferromagnetic ordering we study the time evolution of this initial measure under independent diffusions

pseudo free energies and large deviations for non gibbsian fkg measures a large deviation theorem for the invariant measures of translation invariant attractive interacting particle systems on 0 1 zd is proven in this way a pseudo free energy and pressure is defined for ergodic systems the large deviations property holds with the usual scaling the case of non ergodic systems is also discussed a similar result holds for occupation times the perturbation by an external field is treated

continuous spin mean field models limiting kernels and gibbs properties of local transforms we extend the notion of gibbsianness for mean field systems to the setup of general possibly continuous local state spaces we investigate the gibbs properties of systems arising from an initial mean field gibbs measure by application of given local transition kernels this generalizes previous case studies made for spins taking finitely many values to the first step in the direction to a general theory containing the following parts 1 a formula for the limiting conditional probability distributions of the transformed system it holds both in the gibbs and in the non gibbs regime and invokes a minimization problem for a constrained rate function 2 a criterion for gibbsianness of the transformed system for initial lipschitz hamiltonians involving concentration properties of the transition kernels and 3 a continuity estimate for the single site conditional distributions of the transformed system while 2 and 3 have provable lattice counterparts the characterization of 1 is stronger in m

variational description of gibbs non gibbs dynamical transitions for the curie weiss model we perform a detailed study of gibbs non gibbs transitions for the curie weiss model subject to independent spin flip dynamics infinite temperature dynamics we show that in this setup the program outlined in van enter fernandez den hollander and redig 3 can be fully completed namely that gibbs non gibbs transitions are equivalent to bifurcations in the set of global minima of the large deviation rate function for the trajectories of the magnetization conditioned on their endpoint as a consequence we show that the time evolved model is non gibbs if and only if this set is not a singleton for some value of the final magnetization a detailed description of the possible scenarios of bifurcation is given leading to a full characterization of passages from gibbs to non gibbs and vice versa with sharp transition times under the dynamics gibbsianness can be lost and can be recovered our analysis expands the work of ermolaev and kulske 7 who considered zero mag netic field and finite temperature spin flip dynamics we consider both zero and non zero magnetic field but restricted to infinite temperature spin flip dynamics our results re veal an interesting dependence on the interaction parameters including the presence of forbidden regions for the optimal trajectories and the possible occurrence of overshoots and undershoots in the optimal trajectories the numerical plots provided are obtained with the help of mathematica

two connections between random systems and non gibbsian measures in this contribution we discuss the role disordered or random systems have played in the study of non gibbsian measures this role has two main aspects the distinction between which has not always been fully clear 1 from disordered systems disordered systems can be used as a tool analogies with as well as results and methods from the study of random systems can be employed to investigate non gibbsian properties of a variety of measures of physical and mathematical interest 2 of disordered systems non gibbsianness is a property of various joint measures describing quenched disordered systems we discuss and review this distinction and a number of results related to these issues moreover we discuss the mean field version of the non gibbsian property and present some ideas how a kac limit approach might connect the finite range and the mean field non gibbsian properties

a measure of variance for hierarchical nominal attributes the need for measuring the dispersion of nominal categorical attributes appears in several applications like clustering or data anonymization for a nominal attribute whose categories can be hierarchically classified a measure of the variance of a sample drawn from that attribute is proposed which takes the attribute s hierarchy into account the new measure is the reciprocal of consanguinity the less related the nominal categories in the sample the higher the measured variance for non hierarchical nominal attributes the proposed measure yields results consistent with previous diversity indicators applications of the new nominal variance measure to economic diversity measurement and data anonymization are also discussed

critical behavior of the two dimensional spin diluted ising model via the equilibrium ensemble approach the equilibrium ensemble approach to disordered systems is used to investigate the critical behavior of the two dimensional ising model in the presence of quenched random site dilution the numerical transfer matrix technique in semi infinite strips of finite width together with phenomenological renormalization and conformal invariance is particularly suited to putting the equilibrium ensemble approach to work a method by which to extract with great precision the critical temperature of the model is proposed and applied a more systematic finite size scaling analysis than in previous numerical studies has been performed a parallel investigation along the lines of the two main scenarios currently under discussion namely the logarithmic corrections scenario with critical exponents fixed in the ising universality class versus the weak universality scenario critical exponents varying with the degree of disorder is carried out in interpreting our data maximum care is constantly taken to be open to both possibilities a critical discussion shows that an unambiguous discrimination between the two scenarios is still not possible on the basis of the available finite size data s1063 651x 99 16110 5

interfaces and typical gibbs configurations for one dimensional kac potentials we consider a one dimensional ising spin system with a ferromagnetic kac potential j r j having compact support we study the system in the limit 0 below the lebowitz penrose critical temperature where there are two distinct thermodynamic phases with different magnetizations we prove that the empirical spin average in blocks of size 1 for any positive converges as 0 to one of the two thermodynamic magnetizations uniformly in the intervals of size p for any given positivep 1 we then show that the intervals where the magnetization is approximately constant have lengths of the order of exp c 1 c 0 and that when normalized they converge to independent variables with exponential distribution we show this by proving large deviation estimates and applying the ventsel and friedlin methods to gibbs random fields finally if the temperature is low enough we characterize the interface namely the typical magnetization pattern in the region connecting the two phases

the low temperature phase of kac ising models we analyze the low temperature phase of ferromagnetic kax ising models in dimensionsd 2 we show that if the range of interactions is 1 then two disjoint translation invariant gibbs states exist if the inverse temperature satisfies 1 n where d 1 2d 2 d 1 for any e 0 the proof involves the blocking procedure usual for kac models and also a contour representation for the resulting long range almost continuous spin system which is suitable for the use of a variant of the peierls argument

tightness of probabilities on c lbrack 0 1 rbrack mathscr y and d lbrack 0 1 rbrack mathscr y let c mathscr i c lbrack 0 1 rbrack mathscr i be the space of all continuous mappings of lbrack 0 1 rbrack to mathscr i where mathscr l is the topological dual of the schwartz space mathscr i of all rapidly decreasing functions let c be the banach space of all continuous functions on lbrack 0 1 rbrack for each varphi in mathscr i pi varphi is defined by pi varphi x in c mathscr i rightarrow x varphi in c given a sequence of probability measures p n on c mathscr i such that for each varphi in mathscr i p n pi 1 varphi is tight in c we prove that p n itself is tight in c mathscr i a similar result is proved for the space of all right continuous mappings of lbrack 0 1 rbrack to mathscr i

diffusions conditionnelles ii generateur conditionnel application au filtrage abstract this paper is a continuation of diffusions conditionelles i if x t z t is a two component diffusion process it is shown that under appropriate conditions the process x t t t given z s s t is a nonhomogeneous strong markov process whose generator is explicitly found by using the theory of stochastic flows the filtering equation is reduced to an ordinary partial differential equation

long range order in nonequilibrium systems of interacting brownian linear oscillators long range order lro is established with the help of a generalized peierls argument for non equilibrium lattice systems of one dimensional linear interacting oscillators whose equation of motion for a finite number of them is the smolouchowski equation for the density of a probability distribution interaction is mediated through the pair nearest neighbor quadratic translation invariant potential the initial density is gibbsian with a potential energy satisfying the ruelle superstability and regularity conditions

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

percolation path large deviations and weakly gibbs states we present a unified approach to establishing the gibbsian character of a wide class of non gibbsian states arising in the renormalisation group theory inside the realm of the pirogov sinai theory for lattice spin systems we prove that rg transformations applied to low temperature phases give rise to weakly gibbsian measures in other words we show that the griffiths pearce israel scenario of rg pathologies is carried by atypical configurations the renormalized measures are described by an effective interaction with relative energies well defined on a full measure set of configurations in this way we complete the first part of the dobrushin restoration program to give a gibbsian description to non gibbsian states a disagreement percolation estimate is used in the proof to bound the decay of quenched correlations through which the interaction potential is constructed the percolation is controlled via a novel type of pathwise large deviation theory

majority rule at low temperatures on the square and triangular lattices we consider the majority rule renormalization group transformation applied to nearest neighbor ising models for the square lattice with 2 by 2 blocks we prove that if the temperature is sufficiently low then the transformation is not defined we use the methods of van enter fewrnandez and sokal who proved the renormalized measure is not gibbsian for 7 by 7 blocks if the temperature is too low for the triangular lattice we prove that a zero temperature majorityrule transformation may be defined the resulting renormalized hamiltonian is local with 14 different types of interactions

decay of correlations under dobrushin s uniqueness condition and its applications an estimate on the correlation of functionals of gibbs fields satisfying dobrushin s uniqueness condition is given as a consequence a result of gross saying that the truncated pair correlation function decays in the same weighted summability sense as the potential can be extended to the whole dobrushin uniqueness region applications to the central limit theorem and the second derivative of the pressure are also given

spin flip dynamics of the curie weiss model loss of gibbsianness with possibly broken symmetry we study the conditional probabilities of the curie weiss ising model in vanishing external field under a symmetric independent stochastic spin flip dynamics and discuss their set of points of discontinuity bad points we exhibit a complete analysis of the transition between gibbsian and non gibbsian behavior as a function of time extending the results for the corresponding lattice model where only partial answers can be obtained for initial temperature beta 1 geq 1 we prove that the time evolved measure is always gibbsian for frac 2 3 leq beta 1 1 the time evolved measure loses its gibbsian character at a sharp transition time for beta 1 frac 2 3 we observe the new phenomenon of symmetry breaking in the set of points of discontinuity bad points corresponding to non zero spin average appear at a sharp transition time and give rise to biased non gibbsianness of the time evolved measure these bad points become neutral at a later transition time while the measure stays non gibbs in our proof we give a detailed description of the phase diagram of a curie weiss random field ising model with possibly non symmetric random field distribution based on bifurcation analysis

a gibbs variational principle in space time for infinite dimensional diffusions we show that the set of stationary weak solutions for a class of infinite dimensional stochastic differential equations coincides with the set of shift invariant space time gibbs fields for a certain potential the key step consists in proving the gibbs variational principle for space time gibbs fields

on the central limit theorem for stationary mixing random fields a simple proof of a central limit theorem for stationary random fields under mixing conditions is given generalizing some results obtained by more complicated methods e g bernstein s method

large deviation estimates for a conditional probability distribution applications to random interaction gibbs measures let x i y i d be independent identically distributed random variables with arbitrary distribution we show that for almost every y i i the conditional law of the empirical field given y i i satisfies to large deviation inequalities this applies to the study of gibbs measures with random interaction in the case of some mean field models as well as of short range summable interaction we show that the pressure is nonrandom and is given by a variational formula these random gibbs measures have the same large deviation rate which does not depend on the particular realization of the interaction their local behaviour is described in terms of conditional probabilities given the interaction of solutions to the variational formula

on the ising model with random boundary condition the infinite volume limit behavior of the 2d ising model under possibly strong random boundary conditions is studied the model exhibits chaotic size dependence at low temperatures and we prove that the and phases are the only almost sure limit gibbs measures assuming that the limit is taken along a sparse enough sequence of squares in particular we provide an argument to show that in a sufficiently large volume a typical spin configuration under a typical boundary condition contains no interfaces in order to exclude mixtures as possible limit points a detailed multi scale contour analysis is performed

generalized probabilistic scale space for image restoration a novel generalized sampling based probabilistic scale space theory is proposed for image restoration we explore extending the definition of scale space to better account for both noise and observation models which is important for producing accurately restored images a new class of scale space realizations based on sampling and probability theory is introduced to realize this extended definition in the context of image restoration experimental results using 2 d images show that generalized sampling based probabilistic scale space theory can be used to produce more accurate restored images when compared with state of the art scale space formulations particularly under situations characterized by low signal to noise ratios and image degradation