concentration of measure inequalities for markov chains and phi mixing processes we prove concentration inequalities for some classes of markov chains and f mixing processes with constants independent of the size of the sample that extend the inequalities for product measures of talagrand the method is based on information inequalities put forward by marton in case of contracting markov chains using a simple duality argument on entropy our results also include the family of logarithmic sobolev inequalities for convex functions applications to bounds on supremum of dependent empirical processes complete this work

numerical solution of an etching problem abstract after a short description of the chemical background of etching processes a simplified mathematical model is formulated this contains two parameters that determine the etching profile two approaches for obtaining a numerical solution are discussed first the problem in terms of a variational inequality is reformulated and some new results on its numerical solution are presented then how the problem can be solved by means of a moving grid method as well is explored results obtained by both methods are presented and compared the latter method is applicable to a much wider class of boundary conditions but the variational inequality approach seems attractive for other reasons finally this work is compared with some experimental results

how should one define a weak crystal we compare two proposals for the study of positional long range order one in terms of the spectrum of the translation operator the other in terms of the fourier spectrum we point out that only the first one allows for the consideration of molecular as opposed to atomic weakly periodic structures we illustrate this point on the thue morse system

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

diffractive point sets with entropy after a brief historical survey this paper introduces the notion of entropic model sets cut and project sets and more generally the notion of diffractive point sets with entropy such sets may be thought of as generalizations of lattice gases we show that taking the site occupation of a model set stochastically results with probabilistic certainty in well defined diffractive properties augmented by a constant diffuse background we discuss both the case of independent but identically distributed i i d random variables and that of independent but different i e site dependent random variables several examples are shown

diffraction of stochastic point sets explicitly computable examples stochastic point processes relevant to the theory of long range aperiodic order are considered that display diffraction spectra of mixed type with special emphasis on explicitly computable cases together with a unified approach of reasonable generality the latter is based on the classical theory of point processes and the palm distribution several pairs of autocorrelation and diffraction measures are discussed which show a duality structure analogous to that of the poisson summation formula for lattice dirac combs

comparison theorems for gibbs measures the dobrushin comparison theorem is a powerful tool to bound the difference between the marginals of high dimensional probability distributions in terms of their local specifications originally introduced to prove uniqueness and decay of correlations of gibbs measures it has been widely used in statistical mechanics as well as in the analysis of algorithms on random fields and interacting markov chains however the classical comparison theorem requires validity of the dobrushin uniqueness criterion essentially restricting its applicability in most models to a small subset of the natural parameter space in this paper we develop generalized dobrushin comparison theorems in terms of influences between blocks of sites in the spirit of dobrushin shlosman and weitz that substantially extend the range of applicability of the classical comparison theorem our proofs are based on the analysis of an associated family of markov chains we develop in detail an application of our main results to the analysis of sequential monte carlo algorithms for filtering in high dimension

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

vertex representations for the ν 1 toroidal lie algebra of type bl abstract we give vertex operator constructions for the toroidal lie algebra of type b l 1 l 2 we prove that the subquotients of the modules are completely reducible and give the irreducible decomposition

vertex operators for non simply laced algebras a vertex operator construction is given for the level one representations of the affine kac moody algebras associated with non simply laced finite dimensional lie algebras using free boson and interacting fermion fields the fermion fields are constructed explicitly and a detailed discussion is given of the theory of the cocycles necessary for this and other vertex operator constructions the construction is related in detail to the folding of dynkin diagrams and a generalisation of it yields freudenthal s magic square

measure concentration for a class of random processes let x x i i be a stationary random process with a countable alphabet and distribution q let q x k 0 denote the conditional distribution of x x 1 x 2 x n given the k length past write d 1 x 1 0 if 1 x 1 and d 1 x 1 1 otherwise we say that the process x admits a joining with finite distance u if for any two past sequences k 0 k 1 0 and x k 0 x k 1 x 0 there is a joining of q k 0 and q x k 0 say dist 0 x 0 k 0 x k 0 such that the main result of this paper is the following inequality for processes that admit a joining with finite distance

majorizing measures the generic chaining majorizing measures provide bounds for the supremum of stochastic processes they represent the most general possible form of the chaining argument going back to kolmogorov majorizing measures arose from the theory of gaussian processes but they now have applications far beyond this setting the fundamental question is the construction of these measures this paper focuses on the tools that have been developed for this purpose and in particular the use of geometric ideas applications are given to several natural problems where entropy methods are powerless

the equivalence of the logarithmic sobolev inequality and the dobrushin shlosman mixing condition given a finite range lattice gas with a compact continuous spin space it is shown cf theorem 1 2 that a uniform logarithmic sobolev inequality cf 1 4 holds if and only if the dobrushin shlosman mixing condition cf 1 5 holds as a consequence of our considerations we also show cf theorems 3 2 and 3 6 that these conditions are equivalent to a statement about the uniform rate at which the associated glauber dynamics tends to equilibrium in this same direction we show cf theorem 3 19 that these ideas lead to a surprisingly strong large deviation principle for the occupation time distribution of the glauber dynamics

universal bounds on the selfaveraging of random diffraction measures we consider diffraction at random point scatterers on general discrete point sets in restricted to a finite volume we allow for random amplitudes and random dislocations of the scatterers we investigate the speed of convergence of the random scattering measures applied to an observable towards its mean when the finite volume tends to infinity we give an explicit universal large deviation upper bound that is exponential in the number of scatterers the rate is given in terms of a universal function that depends on the point set only through the minimal distance between points and on the observable only through a suitable sobolev norm our proof uses a cluster expansion and also provides a central limit theorem

on diffraction by aperiodic structures this paper gives a rigorous treatment of some aspects of diffraction by aperiodic structures such as quasicrystals it analyses diffraction in the limit of the infinite system through an appropriately defined autocorrelation the main results are a justification of the standard way of calculating the diffraction spectrum of tilings obtained by the projection method and a proof of a variation on a conjecture by bombieri and taylor

concentration inequalities for random fields via coupling we present a new and simple approach to concentration inequalities in the context of dependent random processes and random fields our method is based on coupling and does not use information inequalities in case one has a uniform control on the coupling one obtains exponential concentration inequalities if such a uniform control is no more possible then one obtains polynomial or stretched exponential concentration inequalities our abstract results apply to gibbs random fields both at high and low temperatures and in particular to the low temperature ising model which is a concrete example of non uniformity of the coupling

a new look at independence the concentration of measure phenomenon in product spaces is a farreaching abstract generalization of the classical exponential inequalities for sums of independent random variables we attempt to explain in the simplest possible terms the basic concepts underlying this phenomenon the basic method to prove concentration inequalities and the meaning of several of the most useful inequalities

close packed dimers on the line diffraction versus dynamical spectrum the translation action of d on a translation bounded measure leads to an interesting class of dynamical systems with a rather rich spectral theory in general the diffraction spectrum of which is the carrier of the diffraction measure lives on a subset of the dynamical spectrum it is known that under some mild assumptions a pure point diffraction spectrum implies a pure point dynamical spectrum the opposite implication always being true for other systems the diffraction spectrum can be a proper subset of the dynamical spectrum as was pointed out for the thue morse sequence with singular continuous diffraction by van enter and mi kisz j stat phys 66 1147 1153 1992 here we construct a random system of close packed dimers on the line that have some underlying long range periodic order as well and display the same type of phenomenon for a system with absolutely continuous spectrum an interpretation in terms of atomic versus molecular spectrum suggests a way to come to a more general correspondence between these two types of spectra

large deviation principles for euclidean functionals and other nearly additive processes we prove a large deviation principle for a process indexed by cubes of the multidimensional integer lattice or euclidean space under approximate additivity and regularity hypotheses the rate function is the convex dual of the limiting logarithmic moment generating function in some applications the rate function can be expressed in terms of relative entropy the general result applies to processes in euclidean combinatorial optimization statistical mechanics and computational geometry examples include the length of the minimal tour the traveling salesman problem the length of the minimal matching graph the length of the minimal spanning tree the length of the k nearest neighbors graph and the free energy of a short range spin glass model

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

weighted dirac combs with pure point diffraction a class of translation bounded complex measures which have the form of weighted dirac combs on locally compact abelian groups is investigated given such a dirac comb we are interested in its diffraction spectrum which emerges as the fourier transform of the autocorrelation measure we present a sufficient set of conditions to ensure that the diffraction measure is a pure point measure simultaneously we establish a natural link to the cut and project formalism and to the theory of almost periodic measures our conditions are general enough to cover the known theory of model sets but also to include examples such as the visible lattice points

coupling concentration inequalities and stochastic dynamics in the context of interacting particle systems we study the influence of the action of the semigroup on the concentration property of lipschitz functions as an application this gives a new approach to estimate the relaxation speed to equilibrium of interacting particle systems we illustrate our approach in a variety of examples for which we obtain several new results with short and nontechnical proofs these examples include the symmetric and asymmetric exclusion processes and high temperature spin flip dynamics glauber dynamics we also give a new proof of the poincare inequality based on coupling in the context of one dimensional gibbs measures in particular we cover the case of polynomially decaying potentials where the log sobolev inequality does not hold

absence of singular continuous diffraction for discrete multi component particle models particle models with finitely many types of particles are considered both on d and on discrete point sets of finite local complexity such sets include many standard examples of aperiodic order such as model sets or certain substitution systems the particle gas is defined by an interaction potential and a corresponding gibbs measure under some reasonable conditions on the underlying point set and the potential we show that the corresponding diffraction measure almost surely exists and consists of a pure point part and an absolutely continuous part with continuous density in particular no singular continuous part is present

vertex operators of g2 1 and bl 1 the vertex operator construction is given for the level one representations of the affine kac moody algebras g2 1 and bl 1

concentration inequalities for functions of gibbs fields with application to diffraction and random gibbs measures we derive useful general concentration inequalities for functions of gibbs fields in the uniqueness regime we also consider expectations of random gibbs measures that depend on an additional disorder field and prove concentration w r t the disorder field both fields are assumed to be in the uniqueness regime allowing in particular for non independent disorder fields the modification of the bounds compared to the case of an independent field can be expressed in terms of constants that resemble the dobrushin contraction coefficient and are explicitly computable on the basis of these inequalities we obtain bounds on the deviation of a diffraction pattern created by random scatterers located on a general discrete point set in euclidean space restricted to a finite volume here we also allow for thermal dislocations of the scatterers around their equilibrium positions extending recent results for independent scatterers we give a universal upper bound on the probability of a deviation of the random scattering measures applied to an observable from its mean the bound is exponential in the number of scatterers with a rate that involves only the minimal distance between points in the point set