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Non) Gibbsianness and phase transitions in random lattice spin models
 Markov Proc. Relat. Fields
, 1999
"... Abstract: We consider disordered lattice spin models with finite volume Gibbs measures µΛ[η](dσ). Here σ denotes a lattice spinvariable and η a lattice random variable with product distribution IP describing the disorder of the model. We ask: When will the joint measures lim Λ↑Z d IP(dη)µΛ[η](dσ) b ..."
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Cited by 25 (12 self)
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Abstract: We consider disordered lattice spin models with finite volume Gibbs measures µΛ[η](dσ). Here σ denotes a lattice spinvariable and η a lattice random variable with product distribution IP describing the disorder of the model. We ask: When will the joint measures lim Λ↑Z d IP(dη)µΛ[η](dσ) be [non] Gibbsian measures on the product of spinspace and disorderspace? We obtain general criteria for both Gibbsianness and nonGibbsianness providing an interesting link between phase transitions at a fixed random configuration and Gibbsianness in product space: Loosely speaking, a phase transition can lead to nonGibbsianness, (only) if it can be observed on the spinobservable conjugate to the independent disorder variables. Our main specific example is the random field Ising model in any dimension for which we show almost sure [almost sure non] Gibbsianness for the single [multi] phase region. We also discuss models with disordered couplings, including spinglasses and ferromagnets, where various mechanisms are responsible for [non] Gibbsianness.
Weakly Gibbsian representations for joint measures of quenched lattice spin models, Probab
 Th. Relat. Fields
, 2001
"... Abstract: Can the joint measures of quenched disordered lattice spin models (with finite range) on the product of spinspace and disorderspace be represented as (suitably generalized) Gibbs measures of an “annealed system”? We prove that there is always a potential (depending on both spin and diso ..."
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Cited by 19 (10 self)
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Abstract: Can the joint measures of quenched disordered lattice spin models (with finite range) on the product of spinspace and disorderspace 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.
A largedeviation view on dynamical gibbsnongibbs transitions
 Moscow Mathematical Journal
"... Abstract. We develop a spacetime largedeviation point of view on GibbsnonGibbs transitions in spin systems subject to a stochastic spinflip dynamics. Using the general theory for large deviations of functionals of Markov processes outlined in a recent book by Feng and Kurtz, we show that the tr ..."
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Cited by 15 (5 self)
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Abstract. We develop a spacetime largedeviation point of view on GibbsnonGibbs transitions in spin systems subject to a stochastic spinflip dynamics. Using the general theory for large deviations of functionals of Markov processes outlined in a recent book by Feng and Kurtz, we show that the trajectory under the spinflip dynamics of the empirical measure of the spins in a large block in Z d satisfies a large deviation principle in the limit as the block size tends to infinity. The associated rate function can be computed as the action functional of a Lagrangian that is the Legendre transform of a certain nonlinear generator, playing a role analogous to the momentgenerating function in the GärtnerEllis theorem of large deviation theory when this is applied to finitedimensional Markov processes. This rate function is used to define the notion of "bad empirical measures", which are the discontinuity points of the optimal trajectories (i.e., the trajectories minimizing the rate function) given the empirical measure at the end of the trajectory. The dynamical GibbsnonGibbs transitions are linked to the occurrence of bad empirical measures: for short times no bad empirical measures occur, while for intermediate and large times bad empirical measures are possible. A future research program is proposed to classify the various possible scenarios behind this crossover, which we refer to as a "natureversusnurture" transition. 2000 Math. Subj. Class. Primary: 60F10, 60G60, 60K35; Secondary: 82B26, 82C22.
Relative entropy and variational properties of generalized Gibbsian measures
, 2004
"... 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 obt ..."
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Cited by 14 (8 self)
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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 translationinvariant 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. 1. Introduction. Since the discovery of the Griffiths–Pearce singularities of renormalization group transformations [8, 28], a challenging question has been whether the classical Gibbs formalism can be extended in such a way
Bounded Fluctuations and Translation Symmetry Breaking in OneDimensional Particle Systems
, 2001
"... Abstract We present general results for onedimensional systems of point charges (signed point measures) on the line with a translation invariant distribution µ for which the variance of the total charge in an interval is uniformly bounded (instead of increasing with the interval length). When the c ..."
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Cited by 9 (3 self)
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Abstract We present general results for onedimensional systems of point charges (signed point measures) on the line with a translation invariant distribution µ for which the variance of the total charge in an interval is uniformly bounded (instead of increasing with the interval length). When the charges are restricted to multiples of a common unit, and their average charge density does not vanish, then the boundedness of the variance implies translationsymmetry breaking — in the sense that there exists a function of the charge configuration that is nontrivially periodic under translations — and hence that µ is not “mixing. ” Analogous results are formulated also for one dimensional lattice systems under some constraints on the values of the charges at the lattice sites and their averages. The general results apply to onedimensional Coulomb systems, and to certain spin chains, putting on common grounds different instances of symmetry breaking encountered there. Dedicated to the memory of J. M. (Quin) Luttinger, a master of onedimensional systems
The Restriction of the Ising Model to a Layer
, 1998
"... We discuss the status of recent Gibbsian descriptions of the restriction (projection) of the Ising phases to a layer. We concentrate on the projection of the twodimensional low temperature Ising phases for which we prove a variational principle. ..."
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Cited by 8 (7 self)
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We discuss the status of recent Gibbsian descriptions of the restriction (projection) of the Ising phases to a layer. We concentrate on the projection of the twodimensional low temperature Ising phases for which we prove a variational principle.
Variational Principle and Almost Quasilocality for Renormalized Measures.
"... We restore part of the thermodynamic formalism for some renormalized measures that are known to be nonGibbsian. We first point out that a recent theory due to Pfister implies that for blocktransformed measures free energy and relative entropy densities exist and are conjugated convex functiona ..."
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Cited by 7 (5 self)
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We restore part of the thermodynamic formalism for some renormalized measures that are known to be nonGibbsian. We first point out that a recent theory due to Pfister implies that for blocktransformed measures free energy and relative entropy densities exist and are conjugated convex functionals. We then determine a necessary and sufficient condition for consistency with a specification that is quasilocal in a fixed direction. As corollaries we obtain consistency results for models with FKG monotonicity and for models with appropriate "continuity rates". For (noisy) decimations or projections of the Ising model these results imply almost quasilocality of the decimated "+" and "\Gamma" measures.