Aernout van enter

University of Groningen | RUG · Johann Bernoulli Institute for Mathematics and Computer Science (JBI)

We show how decimated Gibbs measures which have an unbroken continuous symmetry due to the Mermin-Wagner theorem, although their discrete equivalents have a phase transition, still can become non-Gibbsian. The mechanism rests on the occurrence of a spin-flop transition with a broken discrete symmetry, once the model is constrained by the decimated...

We extend proofs of non-Gibbsianness of decimated Gibbs measures at low temperatures to include long-range as well as vector-spin interactions. Our main tools consist in a two-dimensional use of “equivalence of boundary conditions” in the long-range case and an extension of global specifications for two-dimensional vector spins.

Lattice systems form a widely studied class of models originating in statistical mechanics, and consisting of single‐site observables (functions or operators) describing “spins,” living on infinite lattices. They have found applications in various other fields, in physics (e.g., field theory), mathematics (dynamical systems and probability theory),...

We extend proofs of non-Gibbsianness of decimated Gibbs measures at low temperatures to include long-range, as well as vector-spin interactions. Our main tools consist in a two-dimensional use of ``Equivalence of boundary conditions'' in the long-range case and an extension of Global specifications for two-dimensional vector spins.

We study the one-dimensional projection of the extremal Gibbs measures of the two-dimensional Ising model, the "Schonmann projection". These measures are known to be non-Gibbsian at low temperatures, since their conditional probabilities as a function of the two-sided boundary conditions are not continuous. We prove that they are g-measures, which...

Random boundary conditions are one of the simplest realizations of quenched disorder. They have been used as an illustration of various conceptual issues in the theory of disordered spin systems. Here we review some of these results.

We discuss what ground states for generic interactions look like. We note that a recent result, due to Morris, implies that the behaviour of ground-state measures for generic interactions is similar to that of generic measures. In particular, it follows from his observation that they have singular spectrum and that they are weak mixing, but not mix...

In this review-type paper written at the occasion of the Oberwolfach workshop {\em One-sided vs. Two-sided stochastic processes} (february 22-29, 2020), we discuss and compare Markov properties and generalisations thereof in more directions, as well as weaker forms of conditional dependence, again either in one or more directions. In particular, we...

We discuss what ground states for generic interactions look like. We note that a recent result, due to Morris, implies that the behaviour of ground-state measures for generic interactions is similar to that of generic measures. In particular, it follows from his observation that they have singular spectrum and that they are weak mixing, but not mix...

We construct for the first time examples of non-frustrated, two-body, infinite-range, one-dimensional classical lattice–gas models without periodic ground-state configurations. Ground-state configurations of our models are Sturmian sequences defined by irrational rotations on the circle. We present minimal sets of forbidden patterns which define St...

Random boundary conditions are one of the simplest realizations of quenched disorder. They have been used as an illustration of various conceptual issues in the theory of disordered spin systems. Here we review some of these results.

It is well-known that discrete-time finite-state Markov Chains, which are described by one-sided conditional probabilities which describe a dependence on the past as only dependent on the present, can also be described as one-dimensional Markov Fields, that is, nearest-neighbor Gibbs measures for finite-spin models, which are described by two-sided...

We show that Sturmian sequences defined by irrational rotations on the circle are most-homogeneous. We present minimal sets of forbidden patterns which define them in a unique way. This allows us to construct non-frustrated infinite-range two-body Hamiltonians (augmented by some finite-range interactions) of one-dimensional classical lattice-gas mo...

In this note we study metastability phenomena for a class of long-range Ising models in one-dimension. We prove that, under suitable general conditions, the configuration -1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek}...

It is well-known that discrete-time finite-state Markov Chains, which are described by one-sided conditional probabilities which describe a dependence on the past as only dependent on the present, can also be described as one-dimensional Markov Fields, that is, nearest-neighbour Gibbs measures for finite-spin models, which are described by two-side...

We consider Dyson models, Ising models with slow polynomial decay, at low temperature and show that its Gibbs measures deep in the phase transition region are not $g$-measures. The main ingredient in the proof is the occurrence of an entropic repulsion effect, which follows from the mesoscopic stability of a (single-point) interface for these long-...

We study the critical probability for the metastable phase transition of the two-dimensional anisotropic bootstrap percolation model with $(1,2)$-neighbourhood and threshold $r = 3$. The first order asymptotics for the critical probability were recently determined by the first and second authors. Here we determine the following sharp second and thi...

In this note we study metastability phenomena for a class of long-range Ising models in one-dimension. We prove that, under suitable general conditions, the configuration −1 is the only metastable state and we estimate the mean exit time. Moreover, we illustrate the theory with two examples (exponentially and polynomially decaying interaction) and...

In this note we study metastability phenomena for a class of long-range Ising models in one-dimension. We prove that, under suitable general conditions, the configuration -1 is the only metastable state and we estimate the mean exit time. Moreover, we illustrate the theory with two examples (exponentially and polynomially decaying interaction) and...

We consider the two-dimensional Ising model with long-range pair interactions of the form $J_{xy}\sim|x-y|^{-\alpha}$ with $\alpha>2$, mostly when $J_{xy} \geq 0$. We show that Dobrushin states (i.e. extremal non-translation-invariant Gibbs states selected by mixed $\pm$-boundary conditions) do not exist. We discuss possible extensions of this resu...

In bootstrap percolation, it is known that the critical percolation threshold tends to converge slowly to zero with increasing system size, or, inversely, the critical size diverges fast when the percolation probability goes to zero. To obtain higher-order terms (i.e. sharp and sharper thresholds) for the percolation threshold in general is a hard...

In this paper we study the double transpose extension of the Ruelle transfer operator $\mathscr{L}_{f}$ associated to a general real continuous potential $f\in C(\Omega)$, where $\Omega=E^{\mathbb{N}}$ and $E$ is any compact metric space. For this extension, we prove the existence of non-negative eigenfunctions, in the Banach lattice sense, associa...

We consider ferromagnetic Dyson models which display phase transitions. They are long-range one-dimensional Ising ferromagnets, in which the interaction is given by $J_{x,y} = J(|x-y|)\equiv \frac{1}{|x-y|^{2-\alpha}}$ with $\alpha \in [0, 1)$, in particular, $J(1)=1$. For this class of models one way in which one can prove the phase transition is...

We consider one-dimensional long-range spin models (usually called Dyson models), consisting of Ising ferromagnets with slowly decaying long-range pair potentials of the form $\frac{1}{|i-j|^{\alpha}}$ mainly focusing on the range of slow decays $1 < \alpha \leq 2$. We describe two recent results, one about renormalization and one about the effect...

We consider the ferromagnetic Ising model with spatially dependent external fields on a Cayley tree, and we investigate the conditions for the existence of the phase transition for a class of external fields, asymptotically approaching a homogeneous critical external field. Our results extend earlier results by Rozikov and Ganikhodjaev.

We consider different problems within the general theme of long-range percolation on oriented graphs. Our aim is to settle the so-called truncation question, described as follows. We are given probabilities that certain long-range oriented bonds are open; assuming that the sum of these probabilities is infinite, we ask if the probability of percola...

We study the decimation to a sublattice of half the sites, of the one-dimensional Dyson-Ising ferromagnet with slowly decaying long-range pair interactions of the form $\frac{1}{{|i-j|}^{\alpha}}$, in the phase transition region (1< $\alpha \leq$ 2, and low temperature). We prove non-Gibbsianness of the decimated measure at low enough temperatures...

In this article a multilayer parking system with screening of size n=3 is studied with a focus on the time-dependent particle density. We prove that the asymptotic limit of the particle density increases from an average density of 1/3 on the first layer to the value of (10 - \sqrt 5 )/19 in higher layers.

In bootstrap percolation it is known that the critical percolation threshold tends to converge slowly to zero with increasing system size, or, inversely, the critical size diverges fast when the percolation probability goes to zero. To obtain higher-order terms (that is, sharp and sharper thresholds) for the percolation threshold in general is a ha...

Spatial aperiodicity occurs in various models and material s. Although today the most well-known examples occur in the area of quasicrystals, other applications might also be of interest. Here we discuss some issues related to the notion and occurrence of aperiodic order in equilibrium statistical mechanics. In particular, we consider some spectral...

It is well-known that the dynamical spectrum of an ergodic measure dynamical system is related to the diffraction measure of a typical element of the system. This situation includes ergodic subshifts from symbolic dynamics as well as ergodic Delone dynamical systems, both via suitable embeddings. The connection is rather well understood when the sp...

in this article a multilayer parking system of size n=3 is studied. We prove that the asymptotic limit of the particle density in the center approaches a maximum of 1/2 in higher layers. This means a significant increase of capacity compared to the first layer where this value is 1/3. This is remarkable because the process is solely driven by rando...

Gibbs measures are the main object of study in equilibrium statistical mechanics, and are used in many other contexts, including dynamical systems and ergodic theory, and spatial statistics. However, in a large number of natural instances one encounters measures that are not of Gibbsian form. We present here a number of examples of such non-Gibbsia...

In this paper we analyze several anisotropic bootstrap percolation models in three dimensions. We present the order of magnitude for the metastability threshold for a fairly general class of models. In our proofs we use an adaptation of the technique of dimensional reduction. We find that the order of the metastability threshold is generally determ...

We study a variant of the ferromagnetic Potts model, recently introduced by Tamura, Tanaka and Kawashima, consisting of a ferromagnetic interaction among $q$ "visible" colours along with the presence of $r$ non-interacting "invisible" colours. We introduce a random-cluster representation for the model, for which we prove the existence of a first-or...

In some recent papers by Tamura, Tanaka and Kawashima [arXiv:1102.5475, arXiv:1012.4254], a class of Potts models with "invisible" states was introduced, for which the authors argued by numerical arguments and by a mean-field analysis that a first-order transition occurs. Here we show that the existence of this first-order transition can be proven...

We point out that the claim of strong universality of the paper by Komura and Okabe (2011 J. Phys. A: Math. Theor. 44 015002) is incorrect, as it contradicts known rigorous results.

We strengthen a result from [the second and the third author, Electron. J. Probab. 13, 1307–1344 (2008; Zbl 1190.60096)] on the existence of effective interactions for discretized continuous-spin models. We also point out that such an interaction cannot exist at very low temperatures. Moreover, we compare two ways of discretizing continuous-spin mo...

We point out that the claim of strong universality in the paper J.Phys. A 44, 015002, arXiv:1011.3321 is incorrect, as it contradicts known rigorous results.

The translation action of ℝ d on a translation bounded measure ω leads to an interesting class of dynamical systems, with a rather rich spectral theory. In general, the diffraction spectrum of ω, 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...

Bootstrap percolation models have been extensively studied during the two past decades. In this article, we study the following "anisotropic" bootstrap percolation model: the neighborhood of a point (m,n) is the set \[\{(m+2,n),(m+1,n),(m,n+1),(m-1,n),(m-2,n),(m,n-1)\}.\] At time 0, sites are occupied with probability p. At each time step, sites th...

We compute the Parisi overlap distribution for paperfolding sequences. It turns out to be discrete, and to live on the dyadic rationals. Hence it is a pure point measure whose support is the full interval [-1; +1]. The space of paperfolding sequences has an ultrametric structure. Our example provides an illustration of some properties which were su...

In this paper we study homogeneous Gibbs measures on a Cayley tree, subjected to an infinite-temperature Glauber evolution, and consider their (non-)Gibbsian properties. We show that the intermediate Gibbs state (which in zero field is the free-boundary-condition Gibbs state) behaves different from the plus and the minus state. E.g. at large times,...

We develop a space-time large-deviation point of view on Gibbs-non-Gibbs transitions in spin systems subject to a stochastic spin-flip dynamics. Using the general theory for large deviations of functionals of Markov processes outlined in Feng and Kurtz [11], we show that the trajectory under the spin-flip dynamics of the empirical measure of the sp...

Recently Verdu and Weissman introduced erasure entropies, which are meant to measure the information carried by one or more symbols given all of the remaining symbols in the realization of the random process or field. A natural relation to Gibbs measures has also been observed. In his short note we study this relation further, review a few earlier...

We review some recent developments in the study of Gibbs and non-Gibbs properties of transformed n-vector lattice and mean-field models under various transformations. Also, some new results for the loss and recovery of the Gibbs property of planar rotor models during stochastic time evolution are presented.

We consider planar rotors (XY spins) in $\mathbb{Z}^d$, 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...

We review some of the work on non-Gibbsian states of the last 10 years, emphasizing the developments in which Eurandom played a role.

We study the Gibbsian character of time-evolved planar rotor systems (that is, systems which have two-component, classical XY, spins) on , d>=2, in the transient regime, evolving with stochastic dynamics and starting from an initial Gibbs measure [nu]. We model the system with interacting Brownian diffusions moving on circles. We prove that for sma...

In this note we analyze an anisotropic, two-dimensional bootstrap percolation model introduced by Gravner and Griffeath. We present upper and lower bounds on the finite-size effects. We discuss the similarities with the semi-oriented model introduced by Duarte. Comment: Key words: Bootstrap percolation, anisotropy, finite-size effects

We consider statistical mechanics models of continuous spins in a disordered environment. These models have a natural interpretation as effective interface models. It is well known that without disorder there are no interface Gibbs measures in infinite volume in dimension $d=2$, while there are ``gradient Gibbs measures'' describing an infinite-vol...

We present a class of examples of nearest-neighbour, boubded-spin models, in which the low-temperature Gibbs measures do not converge as the temperature is lowered to zero, in any dimension.

In this note we demonstrate the occurrence of first-order transitions in temperature for some recently introduced generalized XY models, and also point out the connection between them and annealed site-diluted (lattice-gas) continuous-spin models.

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) {\em From} disordered systems: Disordered systems can be used as a tool; analogies with, as well as results and methods from the...

Long-range spin-glass models with random Hamiltonians H = - \mathop åi j J(i,j)|i - j| - a Si SjH = - \mathop \sum \limits_{i j} J(i,j)|i - j|^{ - \alpha } S_i S_j where the J(i,j) are independent, identically distributed random variables with mean IE J(i,j) = 0, satisfying some moment conditions, in some respects behave like nonrandom models wi...

Proceedings of the École d'été de physique des Houches 2005 "Mathematical statistical physics

.In this contribution we discuss the role which incoherent boundaryconditions can play in the study of phase transitions. This is a question ofparticular relevance for the analysis of disordered systems, and in particular ofspin glasses. For the moment our mathematical results only apply to ferromag-netic models which have an exact symmetry between...

In this contribution we discuss the occurrence of first-order transitions in temperature in various short-range lattice models with a rotation symmetry. Such transitions turn out to be widespread under the condition that the interaction potentials are sufficiently nonlinear.

We consider various sufficiently nonlinear vector models of ferromagnets, of nematic liquid crystals and of nonlinear lattice gauge theories with continuous symmetries. We show, employing the method of Reflection Positivity and Chessboard Estimates, that they all exhibit first-order transitions in the temperature, when the nonlinearity parameter is...

In this contribution we discuss the role which incoherent boundary conditions can play in the study of phase transitions. This is a question of particular relevance for the analysis of disordered systems, and in particular of spin glasses. For the moment our mathematical results only apply to ferromagnetic models which have an exact symmetry betwee...

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 pa...

We present a novel approach to establishing the variational principle for Gibbs and generalized (weak and almost) Gibbs states. Limitations of a thermodynamical formalism for generalized Gibbs states will be discussed. A new class of intuitively weak Gibbs measures is introduced, and a typical example is studied. Finally, we present a new example o...

We consider various sufficiently nonlinear sigma models for nematic liquid crystal ordering of RP^{N-1} type and of lattice gauge type with continous symmetries. We rigorously show that they exhibit a first-order transition in the temperature. The result holds in dimension 2 or more for the RP^{N-1} models and in dimension 3 or more for the lattice...

We prove that various SO(n)-invariant n-vector models with interactions which have a deep and narrow enough minimum have a first-order transition in the temperature. The result holds in dimension two or more, and is independent on the nature of the low-temperature phase. Comment: latex

We study a Gaussian Potts-Hopfield model. Whereas for Ising spins and two disorder variables per site the chaotic pair scenario is realized, we find that for q-state Potts spins q (q-1)-tuples occur. Beyond the breaking of a continuous stochastic symmetry, we study the fluctuations and obtain the Newman-Stein metastate description for our model.

We study the nearest-neighbour Ising model with a class of random boundary conditions, chosen from a symmetric i.i.d. distribution. We show for dimensions 4 and higher that almost surely the only limit points for a sequence of increasing cubes are the plus and the minus state. For d=2 and d=3 we prove a similar result for sparse sequences of increa...

We consider Ising-spin systems starting from an initial Gibbs measure ν and evolving under a spin-flip dynamics towards a reversible Gibbs measure μ≠ν. Both ν and μ are assumed to have a translation-invariant finite-range interaction. We study the Gibbsian character of the measure νS(t) at time t and show the following: (1) For all ν and μ, νS(t) i...

The critical behavior of the 2D spin-diluted Ising model is investigated by a new method which combines a grand ensemble approach to disordered systems with phenomenological renormalization. We observe a continuous variation of critical exponents with the density rho of magnetic impurities, respecting, however, weak universality in the sense that e...

We point out that the "disorder potential" employed in the grand ensemble approach is ill-defined in considerable generality. Comment: to appear in Phys. Rev. Lett

In this survey we describe some of the history, the evolution, and the present status of the notion of Gibbs random field.

We consider a spin system on sites of a d-dimensional cubic lattice (d≥2), with the values 0, 1 or -1. It is built over the Bernoulli site percolation model, with spins taking the value 0 on empty sites and values ±1 on occupied sites according to the ferromagnetic Ising model distribution on the occupied clusters. The Hamiltonian corresponds to th...

We point out that the high-q Potts model on a regular lattice at its transition temperature provides an example of a non-robust - in the sense recently proposed by Pemantle and Steif- phase transition.

For classical lattice systems with finite (Ising) spins, we show that the implementation of momentum-space renormalization at the level of Hamiltonians runs into the same type of difficulties as found for real-space transformations: Renormalized Hamiltonians are ill-defined in certain regions of the phase diagram.

We study a ``two-pattern'' Hopfield model with Gaussian disorder. We find that there are infinitely many pure states at low temperatures in this model, and we find that the metastate is supported on an infinity of symmetric pairs of pure states. The origin of this phenomenon is the random breaking of a rotation symmetry of the distribution of the d...

The author discusses some recent results on the absence of phase transitions in one-dimensional spin-glass models with polynomially decaying interactions. The author comments on the probabilistic aspects and on the notion of 'weak uniqueness'.

A class of random-site mean-field Potts models is introduced and solved exactly. The bifurcation properties of the resulting mean-field equations are analysed in detail. Particular emphasis is put on the relation between the solutions and the underlying symmetries of the model. It turns out that, in contrast to the Ising case, the introduction of r...

We review what we have learned about the "Renormalization-Group peculiarities" which were discovered about twenty years ago by Griffiths and Pearce, and which questions they asked are still widely open. We also mention some related developments.

For classical lattice systems with finite (Ising) spins, we show that the implementation of momentum-space renormalization at the level of Hamiltonians runs into the same type of difficulties as found for real-space transformations: Renormalized Hamiltonians are ill-defined in certain regions of the phase diagram. PACS: 64.60Ak, 05.50+q, 02.50Cv 1...

We give some sufficient conditions which guarantee that the entropy density in the thermodynamic limit is equal to the thermodynamic limit of the entropy densities of finite-volume (local) Gibbs states.

We present the first example of an exponentially decaying interaction which gives rise to nonperiodic long-range order at positive temperatures.

An example is presented of a measure on a lattice system which has a measure zero set of points (configurations) where some conditional probability can be discontinuous, but does not become a Gibbs measure under decimation (or other) transformations. We also discuss some related issues.

We present the first example of an exponentially decaying interaction which gives rise to non-periodic long-range order at positive temperatures.

. We discuss the Parisi overlap distribution function for various deterministic systems with uncountably many pure ground states. We show examples of trivial, countably discrete, and continuous distributions. 1 In Parisi's proposed solution for the Sherrington-Kirkpatrick spin-glass model [1, 2, 3] there occurs an overlap distribution p(q) which is...

Examples are presented of block-spin transformations which map the Gibbs measures of the Ising model in two or more dimensions at temperature intervals extending to arbitrarily high temperatures onto non-Gibbsian measures. In this way we provide the first example of this kind of pathology for very high temperatures, and as a corollary also the firs...