Christof Kuelske - Academia.edu (original) (raw)

Papers by Christof Kuelske

Journal of Statistical Physics, Jan 20, 2020

We consider the Widom-Rowlinson model on the lattice Z d in two versions, comparing the cases of ... more We consider the Widom-Rowlinson model on the lattice Z d in two versions, comparing the cases of a hard-core repulsion and of a soft-core repulsion between particles carrying opposite signs. For both versions we investigate their dynamical Gibbs-non-Gibbs transitions under an independent stochastic symmetric spin-flip dynamics. While both models have a similar phase transition in the high-intensity regime in equilibrium, we show that they behave differently under time-evolution: the time-evolved soft-core model is Gibbs for small times and loses the Gibbs property for large enough times. By contrast, the time-evolved hard-core model loses the Gibbs property immediately, and for asymmetric intensities, shows a transition back to the Gibbsian regime at a sharp transition time. Keywords Widom-Rowlinson model • Gibbs measures • Non-Gibbsian measures • Stochastic dynamics • Dynamical Gibbs-non-Gibbs transitions • Peierls argument • Dobrushin uniqueness • Percolation • Phase transitions Mathematics Subject Classification 82B20 • 82B26 • 82C20 1 Introduction Dynamical Gibbs-non-Gibbs transitions have attracted much attention over the last years. This started from an investigation of the Ising model under a high-temperature Glauber time-evolution on the lattice in [23]. It was found that, in zero external magnetic field, the Gibbs property is lost at a finite transition time, after which the measure continues to be non-Gibbsian. The loss of the Gibbs property is indicated by a very long-range (discontinuous) Communicated by Irene Giardina.

arXiv (Cornell University), Feb 21, 2020

We study gradient models for spins taking values in the integers (or an integer lattice), which i... more We study gradient models for spins taking values in the integers (or an integer lattice), which interact via a general potential depending only on the differences of the spin values at neighboring sites, located on a regular tree with d + 1 neighbors. We first provide general conditions in terms of the relevant p-norms of the associated transfer operator Q which ensure the existence of a countable family of proper Gibbs measures, describing localization at different heights. Next we prove existence of delocalized gradient Gibbs measures, under natural conditions on Q. We show that the two conditions can be fulfilled at the same time, which then implies coexistence of both types of measures for large classes of models including the SOS model, and heavy-tailed models arising for instance for potentials of logarithmic growth.

arXiv (Cornell University), Dec 17, 2020

We consider the soft-core Widom-Rowlinson model for particles with spins and holes, on a Cayley t... more We consider the soft-core Widom-Rowlinson model for particles with spins and holes, on a Cayley tree of order d (which has d + 1 nearest neighbours), depending on repulsion strength β between particles of different signs and on an activity parameter λ for particles. We analyse Gibbsian properties of the time-evolved intermediate Gibbs measure of the static model, under a spin-flip time evolution, in a regime of large repulsion strength β. We first show that there is a dynamical transition, in which the measure becomes non-Gibbsian at large times, independently of the particle activity, for any d ≥ 2. In our second and main result, we also show that for large β and at large times, the measure of the set of bad configurations (discontinuity points) changes from zero to one as the particle activity λ increases, assuming that d ≥ 4. Our proof relies on a general zero-one law for bad configurations on the tree, and the introduction of a set of uniformly bad configurations given in terms of subtree percolation, which we show to become typical at high particle activity.

arXiv (Cornell University), Sep 5, 2016

We consider the continuum Widom-Rowlinson model under independent spin-flip dynamics and investig... more We consider the continuum Widom-Rowlinson model under independent spin-flip dynamics and investigate whether and when the time-evolved point process has an (almost) quasilocal specification (Gibbs-property of the time-evolved measure). Our study provides a first analysis of a Gibbs-non-Gibbs transition for point particles in Euclidean space. We find a picture of loss and recovery, in which even more regularity is lost faster than it is for time-evolved spin models on lattices. We show immediate loss of quasilocality in the percolation regime, with full measure of discontinuity points for any specification. For the color-asymmetric percolating model, there is a transition from this non-a.s. quasilocal regime back to an everywhere Gibbsian regime. At the sharp reentrance time tG > 0 the model is a.s. quasilocal. For the color-symmetric model there is no reentrance. On the constructive side, for all t > tG, we provide everywhere quasilocal specifications for the time-evolved measures and give precise exponential estimates on the influence of boundary condition.

arXiv (Cornell University), Jan 29, 2019

The Widom-Rowlinson model is an equilibrium model for point particles in Euclidean space. It has ... more The Widom-Rowlinson model is an equilibrium model for point particles in Euclidean space. It has a repulsive interaction between particles of different colors, and shows a phase transition at high intensity. Natural versions of the model can moreover be formulated in different geometries: in particular as a lattice system or a mean-field system. We will discuss recent results on dynamical Gibbs-non Gibbs transitions in this context. Main issues will be the possibility or impossibility of an immediate loss of the Gibbs property, and of full-measure discontinuities of the time-evolved models.

HAL (Le Centre pour la Communication Scientifique Directe), Feb 13, 2019

We consider an SOS (solid-on-solid) model, with spin values from the set of all integers, on a Ca... more We consider an SOS (solid-on-solid) model, with spin values from the set of all integers, on a Cayley tree of order k ≥ 2 and are interested in translation-invariant gradient Gibbs measures (GGMs) of the model. Such a measure corresponds to a boundary law (a function defined on vertices of the Cayley tree) satisfying a functional equation. In the ferromagnetic SOS case on the binary tree we find up to five solutions to a class of 4-periodic boundary law equations (in particular, some two periodic ones). We show that these boundary laws define up to four distinct GGMs. Moreover, we construct some 3-periodic boundary laws on the Cayley tree of arbitrary order k ≥ 2, which define GGMs different from the 4-periodic ones.

arXiv (Cornell University), Mar 23, 2014

We continue our study of the full set of translation-invariant splitting Gibbs measures (TISGMs, ... more We continue our study of the full set of translation-invariant splitting Gibbs measures (TISGMs, translation-invariant tree-indexed Markov chains) for the qstate Potts model on a Cayley tree. In our previous work [14] we gave a full description of the TISGMs, and showed in particular that at sufficiently low temperatures their number is 2 q − 1. In this paper we find some regions for the temperature parameter ensuring that a given TISGM is (non-)extreme in the set of all Gibbs measures. In particular we show the existence of a temperature interval for which there are at least 2 q−1 + q extremal TISGMs. For the Cayley tree of order two we give explicit formulae and some numerical values.

arXiv (Cornell University), Mar 23, 2005

We present a new and simple approach to deviation inequalities for non-product measures, i.e., fo... more We present a new and simple approach to deviation inequalities for non-product measures, i.e., for dependent random variables. Our method is based on coupling. We illustrate our abstract results with chains with complete connections and Gibbsian random fields, both at high and low temperature.

arXiv (Cornell University), Apr 26, 1999

Our main specific example is the random field Ising model in any dimension for which we show almo... more 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.

arXiv (Cornell University), Nov 16, 2002

We study the thermodynamic formalism for generalized Gibbs measures, such as renormalization grou... more We study the thermodynamic formalism for generalized Gibbs measures, such as renormalization group transformations of Gibbs measures or joint measures of disordered spin systems. We first show existence of the relative entropy density and obtain a familiar expression in terms of entropy and relative energy for "almost Gibbsian measures" (almost sure continuity of conditional probabilities). We also describe these measures as equilibrium states and establish an extension of the usual variational principle. As a corollary, we obtain a full variational principle for quasilocal measures. 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 loses 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.

arXiv (Cornell University), Apr 16, 2004

We analyze a simple approximation scheme based on the Morita-approach for the example of the mean... more We analyze a simple approximation scheme based on the Morita-approach for the example of the mean field random field Ising model where it is claimed to be exact in some of the physics literature. We show that the approximation scheme is flawed, but it provides a set of equations whose metastable solutions surprisingly yield the correct solution of the model. We explain how the same equations appear in a different way as rigorous consistency equations. We clarify the relation between the validity of their solutions and the almost surely discontinuous behavior of the single-site conditional probabilities.

arXiv (Cornell University), Oct 29, 1999

Can the joint measures of quenched disordered lattice spin models (with finite range) on the prod... more 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.

arXiv (Cornell University), Oct 31, 2003

We study Gibbs properties of the fuzzy Potts model in the mean field case (i.e. on a complete gra... more We study Gibbs properties of the fuzzy Potts model in the mean field case (i.e. on a complete graph) and on trees. For the mean field case, a complete characterization of the set of temperatures for which non-Gibbsianness happens is given. The results for trees are somewhat less explicit, but we do show for general trees that non-Gibbsianness of the fuzzy Potts model happens exactly for those temperatures where the underlying Potts model has multiple Gibbs measures.

We investigate the Gibbs properties of the fuzzy Potts model on the d-dimensional torus with Kac ... more We investigate the Gibbs properties of the fuzzy Potts model on the d-dimensional torus with Kac interaction. We use a variational approach for profiles inspired by that of Fernández, den Hollander and Martínez [17] for their study of the Gibbs-non-Gibbs transitions of a dynamical Kac-Ising model on the torus. As our main result, we show that the mean-field thresholds dividing Gibbsian from non-Gibbsian behavior are sharp in the fuzzy Kac-Potts model. On the way to this result we prove a large deviation principle for color profiles with diluted total mass densities and use monotocity arguments.

arXiv (Cornell University), Apr 12, 2014

We provide an example of a discrete-time Markov process on the threedimensional infinite integer ... more We provide an example of a discrete-time Markov process on the threedimensional infinite integer lattice with Z q-invariant Bernoulli-increments which has as local state space the cyclic group Z q. We show that the system has a unique invariant measure, but remarkably possesses an invariant set of measures on which the dynamics is conjugate to an irrational rotation on the continuous sphere S 1. The update mechanism we construct is exponentially well localized on the lattice.

arXiv (Cornell University), Sep 4, 2001

We consider diffraction at random point scatterers on general discrete point sets in R ν , restri... more We consider diffraction at random point scatterers on general discrete point sets in R ν , 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.

arXiv (Cornell University), Sep 29, 2014

We consider stochastic dynamics of lattice systems with finite local state space, possibly at low... more We consider stochastic dynamics of lattice systems with finite local state space, possibly at low temperature, and possibly non-reversible. We assume the additional regularity properties on the dynamics: a) There is at least one stationary measure which is a Gibbs measure for an absolutely summable potential Φ. b) Zero loss of relative entropy density under dynamics implies the Gibbs property with the same Φ. We prove results on the attractor property of the set of Gibbs measures for Φ: 1. The set of weak limit points of any trajectory of translation-invariant measures contains at least one Gibbs state for Φ. 2. We show that if all elements of a weakly convergent sequence of measures are Gibbs measures for a sequence of some translation-invariant summable potentials with uniform bound, then the limiting measure must be a Gibbs measure for Φ. 3. We give an extension of the second result to trajectories which are allowed to be non-Gibbs, but have a property of asymptotic smallness of discontinuities. An example for this situation is the time evolution from a low temperature Ising measure by weakly dependent spin flips.

arXiv (Cornell University), Feb 14, 2015

We investigate the Gibbs properties of the fuzzy Potts model on the d-dimensional torus with Kac ... more We investigate the Gibbs properties of the fuzzy Potts model on the d-dimensional torus with Kac interaction. We use a variational approach for profiles inspired by that of Fernández, den Hollander and Martínez [18] for their study of the Gibbs-non-Gibbs transitions of a dynamical Kac-Ising model on the torus. As our main result, we show that the mean-field thresholds dividing Gibbsian from non-Gibbsian behavior are sharp in the fuzzy Kac-Potts model with class size unequal two. On the way to this result we prove a large deviation principle for color profiles with diluted total mass densities and use monotocity arguments.

arXiv (Cornell University), Mar 17, 2009

We give a criterion of the form Q(d)c(M) < 1 for the non-reconstructability of tree-indexed q-sta... more We give a criterion of the form Q(d)c(M) < 1 for the non-reconstructability of tree-indexed q-state Markov chains obtained by broadcasting a signal from the root with a given transition matrix M. Here c(M) is an explicit function, which is convex over the set of M 's with a given invariant distribution, that is defined in terms of a q − 1-dimensional variational problem over symmetric entropies. Further Q(d) is the expected number of offspring on the Galton-Watson tree. This result is equivalent to proving the extremality of the free boundary condition Gibbs measure within the corresponding Gibbs-simplex. Our theorem holds for possibly non-reversible M and its proof is based on a general recursion formula for expectations of a symmetrized relative entropy function, which invites their use as a Lyapunov function. In the case of the Potts model, the present theorem reproduces earlier results of the authors, with a simplified proof, in the case of the symmetric Ising model (where the argument becomes similar to the approach of Pemantle and Peres) the method produces the correct reconstruction threshold), in the case of the (strongly) asymmetric Ising model where the Kesten-Stigum bound is known to be not sharp the method provides improved numerical bounds.

arXiv (Cornell University), Dec 22, 1998

We consider the Gibbs-measures of continuous-valued height configurations on the d-dimensional in... more We consider the Gibbs-measures of continuous-valued height configurations on the d-dimensional integer lattice in the presence a weakly disordered potential. The potential is composed of Gaussians having random location and random depth; it becomes periodic under shift of the interface perpendicular to the base-plane for zero disorder. We prove that there exist localized interfaces with probability one in dimensions d ≥ 3 + 1, in a 'low-temperature' regime. The proof extends the method of continuous-to-discrete single-site coarse graining that was previously applied by the author for a double-well potential to the case of a non-compact image space. This allows to utilize parts of the renormalization group analysis developed for the treatment of a contour representation of a related integer-valued SOS-model in [BoK1]. We show that, for a.e. fixed realization of the disorder, the infinite volume Gibbs measures then have a representation as superpositions of massive Gaussian fields with centerings that are distributed according to the infinite volume Gibbs measures of the disordered integer-valued SOS-model with exponentially decaying interactions.

Journal of Statistical Physics, Jan 20, 2020

We consider the Widom-Rowlinson model on the lattice Z d in two versions, comparing the cases of ... more We consider the Widom-Rowlinson model on the lattice Z d in two versions, comparing the cases of a hard-core repulsion and of a soft-core repulsion between particles carrying opposite signs. For both versions we investigate their dynamical Gibbs-non-Gibbs transitions under an independent stochastic symmetric spin-flip dynamics. While both models have a similar phase transition in the high-intensity regime in equilibrium, we show that they behave differently under time-evolution: the time-evolved soft-core model is Gibbs for small times and loses the Gibbs property for large enough times. By contrast, the time-evolved hard-core model loses the Gibbs property immediately, and for asymmetric intensities, shows a transition back to the Gibbsian regime at a sharp transition time. Keywords Widom-Rowlinson model • Gibbs measures • Non-Gibbsian measures • Stochastic dynamics • Dynamical Gibbs-non-Gibbs transitions • Peierls argument • Dobrushin uniqueness • Percolation • Phase transitions Mathematics Subject Classification 82B20 • 82B26 • 82C20 1 Introduction Dynamical Gibbs-non-Gibbs transitions have attracted much attention over the last years. This started from an investigation of the Ising model under a high-temperature Glauber time-evolution on the lattice in [23]. It was found that, in zero external magnetic field, the Gibbs property is lost at a finite transition time, after which the measure continues to be non-Gibbsian. The loss of the Gibbs property is indicated by a very long-range (discontinuous) Communicated by Irene Giardina.

arXiv (Cornell University), Feb 21, 2020

We study gradient models for spins taking values in the integers (or an integer lattice), which i... more We study gradient models for spins taking values in the integers (or an integer lattice), which interact via a general potential depending only on the differences of the spin values at neighboring sites, located on a regular tree with d + 1 neighbors. We first provide general conditions in terms of the relevant p-norms of the associated transfer operator Q which ensure the existence of a countable family of proper Gibbs measures, describing localization at different heights. Next we prove existence of delocalized gradient Gibbs measures, under natural conditions on Q. We show that the two conditions can be fulfilled at the same time, which then implies coexistence of both types of measures for large classes of models including the SOS model, and heavy-tailed models arising for instance for potentials of logarithmic growth.

arXiv (Cornell University), Dec 17, 2020

We consider the soft-core Widom-Rowlinson model for particles with spins and holes, on a Cayley t... more We consider the soft-core Widom-Rowlinson model for particles with spins and holes, on a Cayley tree of order d (which has d + 1 nearest neighbours), depending on repulsion strength β between particles of different signs and on an activity parameter λ for particles. We analyse Gibbsian properties of the time-evolved intermediate Gibbs measure of the static model, under a spin-flip time evolution, in a regime of large repulsion strength β. We first show that there is a dynamical transition, in which the measure becomes non-Gibbsian at large times, independently of the particle activity, for any d ≥ 2. In our second and main result, we also show that for large β and at large times, the measure of the set of bad configurations (discontinuity points) changes from zero to one as the particle activity λ increases, assuming that d ≥ 4. Our proof relies on a general zero-one law for bad configurations on the tree, and the introduction of a set of uniformly bad configurations given in terms of subtree percolation, which we show to become typical at high particle activity.

arXiv (Cornell University), Sep 5, 2016

We consider the continuum Widom-Rowlinson model under independent spin-flip dynamics and investig... more We consider the continuum Widom-Rowlinson model under independent spin-flip dynamics and investigate whether and when the time-evolved point process has an (almost) quasilocal specification (Gibbs-property of the time-evolved measure). Our study provides a first analysis of a Gibbs-non-Gibbs transition for point particles in Euclidean space. We find a picture of loss and recovery, in which even more regularity is lost faster than it is for time-evolved spin models on lattices. We show immediate loss of quasilocality in the percolation regime, with full measure of discontinuity points for any specification. For the color-asymmetric percolating model, there is a transition from this non-a.s. quasilocal regime back to an everywhere Gibbsian regime. At the sharp reentrance time tG > 0 the model is a.s. quasilocal. For the color-symmetric model there is no reentrance. On the constructive side, for all t > tG, we provide everywhere quasilocal specifications for the time-evolved measures and give precise exponential estimates on the influence of boundary condition.

arXiv (Cornell University), Jan 29, 2019

The Widom-Rowlinson model is an equilibrium model for point particles in Euclidean space. It has ... more The Widom-Rowlinson model is an equilibrium model for point particles in Euclidean space. It has a repulsive interaction between particles of different colors, and shows a phase transition at high intensity. Natural versions of the model can moreover be formulated in different geometries: in particular as a lattice system or a mean-field system. We will discuss recent results on dynamical Gibbs-non Gibbs transitions in this context. Main issues will be the possibility or impossibility of an immediate loss of the Gibbs property, and of full-measure discontinuities of the time-evolved models.

HAL (Le Centre pour la Communication Scientifique Directe), Feb 13, 2019

We consider an SOS (solid-on-solid) model, with spin values from the set of all integers, on a Ca... more We consider an SOS (solid-on-solid) model, with spin values from the set of all integers, on a Cayley tree of order k ≥ 2 and are interested in translation-invariant gradient Gibbs measures (GGMs) of the model. Such a measure corresponds to a boundary law (a function defined on vertices of the Cayley tree) satisfying a functional equation. In the ferromagnetic SOS case on the binary tree we find up to five solutions to a class of 4-periodic boundary law equations (in particular, some two periodic ones). We show that these boundary laws define up to four distinct GGMs. Moreover, we construct some 3-periodic boundary laws on the Cayley tree of arbitrary order k ≥ 2, which define GGMs different from the 4-periodic ones.

arXiv (Cornell University), Mar 23, 2014

We continue our study of the full set of translation-invariant splitting Gibbs measures (TISGMs, ... more We continue our study of the full set of translation-invariant splitting Gibbs measures (TISGMs, translation-invariant tree-indexed Markov chains) for the qstate Potts model on a Cayley tree. In our previous work [14] we gave a full description of the TISGMs, and showed in particular that at sufficiently low temperatures their number is 2 q − 1. In this paper we find some regions for the temperature parameter ensuring that a given TISGM is (non-)extreme in the set of all Gibbs measures. In particular we show the existence of a temperature interval for which there are at least 2 q−1 + q extremal TISGMs. For the Cayley tree of order two we give explicit formulae and some numerical values.

arXiv (Cornell University), Mar 23, 2005

We present a new and simple approach to deviation inequalities for non-product measures, i.e., fo... more We present a new and simple approach to deviation inequalities for non-product measures, i.e., for dependent random variables. Our method is based on coupling. We illustrate our abstract results with chains with complete connections and Gibbsian random fields, both at high and low temperature.

arXiv (Cornell University), Apr 26, 1999

Our main specific example is the random field Ising model in any dimension for which we show almo... more 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.

arXiv (Cornell University), Nov 16, 2002

We study the thermodynamic formalism for generalized Gibbs measures, such as renormalization grou... more We study the thermodynamic formalism for generalized Gibbs measures, such as renormalization group transformations of Gibbs measures or joint measures of disordered spin systems. We first show existence of the relative entropy density and obtain a familiar expression in terms of entropy and relative energy for "almost Gibbsian measures" (almost sure continuity of conditional probabilities). We also describe these measures as equilibrium states and establish an extension of the usual variational principle. As a corollary, we obtain a full variational principle for quasilocal measures. 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 loses 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.

arXiv (Cornell University), Apr 16, 2004

We analyze a simple approximation scheme based on the Morita-approach for the example of the mean... more We analyze a simple approximation scheme based on the Morita-approach for the example of the mean field random field Ising model where it is claimed to be exact in some of the physics literature. We show that the approximation scheme is flawed, but it provides a set of equations whose metastable solutions surprisingly yield the correct solution of the model. We explain how the same equations appear in a different way as rigorous consistency equations. We clarify the relation between the validity of their solutions and the almost surely discontinuous behavior of the single-site conditional probabilities.

arXiv (Cornell University), Oct 29, 1999

Can the joint measures of quenched disordered lattice spin models (with finite range) on the prod... more 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.

arXiv (Cornell University), Oct 31, 2003

We study Gibbs properties of the fuzzy Potts model in the mean field case (i.e. on a complete gra... more We study Gibbs properties of the fuzzy Potts model in the mean field case (i.e. on a complete graph) and on trees. For the mean field case, a complete characterization of the set of temperatures for which non-Gibbsianness happens is given. The results for trees are somewhat less explicit, but we do show for general trees that non-Gibbsianness of the fuzzy Potts model happens exactly for those temperatures where the underlying Potts model has multiple Gibbs measures.

We investigate the Gibbs properties of the fuzzy Potts model on the d-dimensional torus with Kac ... more We investigate the Gibbs properties of the fuzzy Potts model on the d-dimensional torus with Kac interaction. We use a variational approach for profiles inspired by that of Fernández, den Hollander and Martínez [17] for their study of the Gibbs-non-Gibbs transitions of a dynamical Kac-Ising model on the torus. As our main result, we show that the mean-field thresholds dividing Gibbsian from non-Gibbsian behavior are sharp in the fuzzy Kac-Potts model. On the way to this result we prove a large deviation principle for color profiles with diluted total mass densities and use monotocity arguments.

arXiv (Cornell University), Apr 12, 2014

We provide an example of a discrete-time Markov process on the threedimensional infinite integer ... more We provide an example of a discrete-time Markov process on the threedimensional infinite integer lattice with Z q-invariant Bernoulli-increments which has as local state space the cyclic group Z q. We show that the system has a unique invariant measure, but remarkably possesses an invariant set of measures on which the dynamics is conjugate to an irrational rotation on the continuous sphere S 1. The update mechanism we construct is exponentially well localized on the lattice.

arXiv (Cornell University), Sep 4, 2001

We consider diffraction at random point scatterers on general discrete point sets in R ν , restri... more We consider diffraction at random point scatterers on general discrete point sets in R ν , 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.

arXiv (Cornell University), Sep 29, 2014

We consider stochastic dynamics of lattice systems with finite local state space, possibly at low... more We consider stochastic dynamics of lattice systems with finite local state space, possibly at low temperature, and possibly non-reversible. We assume the additional regularity properties on the dynamics: a) There is at least one stationary measure which is a Gibbs measure for an absolutely summable potential Φ. b) Zero loss of relative entropy density under dynamics implies the Gibbs property with the same Φ. We prove results on the attractor property of the set of Gibbs measures for Φ: 1. The set of weak limit points of any trajectory of translation-invariant measures contains at least one Gibbs state for Φ. 2. We show that if all elements of a weakly convergent sequence of measures are Gibbs measures for a sequence of some translation-invariant summable potentials with uniform bound, then the limiting measure must be a Gibbs measure for Φ. 3. We give an extension of the second result to trajectories which are allowed to be non-Gibbs, but have a property of asymptotic smallness of discontinuities. An example for this situation is the time evolution from a low temperature Ising measure by weakly dependent spin flips.

arXiv (Cornell University), Feb 14, 2015

We investigate the Gibbs properties of the fuzzy Potts model on the d-dimensional torus with Kac ... more We investigate the Gibbs properties of the fuzzy Potts model on the d-dimensional torus with Kac interaction. We use a variational approach for profiles inspired by that of Fernández, den Hollander and Martínez [18] for their study of the Gibbs-non-Gibbs transitions of a dynamical Kac-Ising model on the torus. As our main result, we show that the mean-field thresholds dividing Gibbsian from non-Gibbsian behavior are sharp in the fuzzy Kac-Potts model with class size unequal two. On the way to this result we prove a large deviation principle for color profiles with diluted total mass densities and use monotocity arguments.

arXiv (Cornell University), Mar 17, 2009

We give a criterion of the form Q(d)c(M) < 1 for the non-reconstructability of tree-indexed q-sta... more We give a criterion of the form Q(d)c(M) < 1 for the non-reconstructability of tree-indexed q-state Markov chains obtained by broadcasting a signal from the root with a given transition matrix M. Here c(M) is an explicit function, which is convex over the set of M 's with a given invariant distribution, that is defined in terms of a q − 1-dimensional variational problem over symmetric entropies. Further Q(d) is the expected number of offspring on the Galton-Watson tree. This result is equivalent to proving the extremality of the free boundary condition Gibbs measure within the corresponding Gibbs-simplex. Our theorem holds for possibly non-reversible M and its proof is based on a general recursion formula for expectations of a symmetrized relative entropy function, which invites their use as a Lyapunov function. In the case of the Potts model, the present theorem reproduces earlier results of the authors, with a simplified proof, in the case of the symmetric Ising model (where the argument becomes similar to the approach of Pemantle and Peres) the method produces the correct reconstruction threshold), in the case of the (strongly) asymmetric Ising model where the Kesten-Stigum bound is known to be not sharp the method provides improved numerical bounds.

arXiv (Cornell University), Dec 22, 1998

We consider the Gibbs-measures of continuous-valued height configurations on the d-dimensional in... more We consider the Gibbs-measures of continuous-valued height configurations on the d-dimensional integer lattice in the presence a weakly disordered potential. The potential is composed of Gaussians having random location and random depth; it becomes periodic under shift of the interface perpendicular to the base-plane for zero disorder. We prove that there exist localized interfaces with probability one in dimensions d ≥ 3 + 1, in a 'low-temperature' regime. The proof extends the method of continuous-to-discrete single-site coarse graining that was previously applied by the author for a double-well potential to the case of a non-compact image space. This allows to utilize parts of the renormalization group analysis developed for the treatment of a contour representation of a related integer-valued SOS-model in [BoK1]. We show that, for a.e. fixed realization of the disorder, the infinite volume Gibbs measures then have a representation as superpositions of massive Gaussian fields with centerings that are distributed according to the infinite volume Gibbs measures of the disordered integer-valued SOS-model with exponentially decaying interactions.