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

Papers by Christof Kuelske

arXiv (Cornell University), Jan 7, 2022

We consider the locally thinned Bernoulli field on Z d , which is the lattice version of the Type... more We consider the locally thinned Bernoulli field on Z d , which is the lattice version of the Type-I Matérn hardcore process in Euclidean space. It is given as the lattice field of occupation variables, obtained as image of an i.i.d. Bernoulli lattice field with occupation probability p, under the map which removes all particles with neighbors, while keeping the isolated particles. We prove that the thinned measure has a Gibbsian representation and provide control on its quasilocal dependence, both in the regime of small p, but also in the regime of large p, where the thinning transformation changes the Bernoulli measure drastically. Our methods rely on Dobrushin uniqueness criteria, disagreement percolation arguments [46], and cluster expansions.

HAL (Le Centre pour la Communication Scientifique Directe), 2007

arXiv (Cornell University), Aug 6, 2013

We analyze a non-reversible mean-field jump dynamics for discrete qvalued rotators and show in pa... more We analyze a non-reversible mean-field jump dynamics for discrete qvalued rotators and show in particular that it exhibits synchronization. The dynamics is the mean-field analogue of the lattice dynamics investigated by the same authors in which provides an example of a non-ergodic interacting particle system on the basis of a mechanism suggested by Maes and Shlosman . Based on the correspondence to an underlying model of continuous rotators via a discretization transformation we show the existence of a locally attractive periodic orbit of rotating measures. We also discuss global attractivity, using a free energy as a Lyapunov function and the linearization of the ODE which describes typical behavior of the empirical distribution vector.

Journal of Statistical Physics, May 20, 2015

We consider the SOS (solid-on-solid) model, with spin values 0, 1, 2, on the Cayley tree of order... more We consider the SOS (solid-on-solid) model, with spin values 0, 1, 2, on the Cayley tree of order two (binary tree). We treat both ferromagnetic and antiferromagnetic coupling, with interactions which are proportional to the absolute value of the spin differences. We present a classification of all translation-invariant phases (splitting Gibbs measures) of the model: We show uniqueness in the case of antiferromagnetic interactions, and existence of up to seven phases in the case of ferromagnetic interactions, where the number of phases depends on the interaction strength. Next we investigate whether these states are extremal or non-extremal in the set of all Gibbs measures, when the coupling strength is varied, whenever they exist. We show that two states are always extremal, two states are always nonextremal, while three of the seven states make transitions between extremality and non-extremality. We provide explicit bounds on those transition values, making use of algebraic properties of the models, and an adaptation of the method of Martinelli, Sinclair, Weitz.

Annales de l'I.H.P, Feb 1, 2023

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.

$Research paper thumbnail of Dynamical Gibbs–non-Gibbs Transitions in the Curie–Weiss Potts Model in the Regime$$\beta <3$$$

Journal of Statistical Physics, Jul 23, 2021

We consider the Curie-Weiss Potts model in zero external field under independent symmetric spin-f... more We consider the Curie-Weiss Potts model in zero external field under independent symmetric spin-flip dynamics. We investigate dynamical Gibbs-non-Gibbs transitions for a range of initial inverse temperatures β < 3, which covers the phase transition point β = 4 log 2 (Ellis and Wang in Stoch Process Appl 35(1): 1990). We show that finitely many types of trajectories of bad empirical measures appear, depending on the parameter β, with a possibility of re-entrance into the Gibbsian regime, of which we provide a full description. Potts model • Curie-Weiss model • Mean-field • Phase transitions • Dynamical Gibbs-non-Gibbs transitions • Sequential Gibbs property • Large deviations • Singularity theory • Butterflies • Beak-to-beak • Umbilics Mathematics Subject Classification 82B20 • 82B26 • 82C20 Communicated by Aernout van Enter.

arXiv (Cornell University), Nov 21, 2014

arXiv (Cornell University), Apr 4, 2007

We consider statistical mechanics models of continuous height effective interfaces in the presenc... more We consider statistical mechanics models of continuous height effective interfaces in the presence of a delta-pinning of strength ε at height zero. There is a detailed mathematical understanding of the depinning transition in 2 dimensions without disorder. Then the variance of the interface height w.r.t. the Gibbs measure stays bounded uniformly in the volume for ε > 0 and diverges like | log ε| for ε ↓ 0 How does the presence of a quenched disorder term in the Hamiltonian modify this transition? We show that an arbitarily weak random field term is enough to beat an arbitrarily strong delta-pinning in 2 dimensions and will cause delocalization. The proof is based on a rigorous lower bound for the overlap between local magnetizations and random fields in finite volume. In 2 dimensions it implies growth faster than the volume which is a contradiction to localization. We also derive a simple complementary inequality which shows that in higher dimensions the fraction of pinned sites converges to one with ε ↑ ∞.

arXiv (Cornell University), Sep 23, 2004

We consider a specific continuous-spin Gibbs distribution µ t=0 for a double-well potential that ... more We consider a specific continuous-spin Gibbs distribution µ t=0 for a double-well potential that allows for ferromagnetic ordering. We study the time-evolution of this initial measure under independent diffusions. For 'high temperature' initial measures we prove that the time-evoved measure µ t is Gibbsian for all t. For 'low temperature' initial measures we prove that µ t stays Gibbsian for small enough times t, but loses its Gibbsian character for large enough t. In contrast to the analogous situation for discrete-spin Gibbs measures, there is no recovery of the Gibbs property for large t in the presence of a non-vanishing external magnetic field. All of our results hold for any dimension d ≥ 2. This example suggests more generally that time-evolved continuous-spin models tend to be non-Gibbsian more easily than their discrete-spin counterparts.

arXiv (Cornell University), Feb 23, 2021

We provide an existence theory for gradient Gibbs measures for Z-valued spin models on regular tr... more We provide an existence theory for gradient Gibbs measures for Z-valued spin models on regular trees which are not invariant under translations of the tree, assuming only summability of the transfer operator. The gradient states we obtain are delocalized. The construction we provide for them starts from a two-layer hidden Markov model representation in a setup which is not invariant under tree-automorphisms, involving internal q-spin models. The proofs of existence and lack of translation invariance of infinite-volume gradient states are based on properties of the local pseudo-unstable manifold of the corresponding discrete dynamical systems of these internal models, around the free state, at large q.

Research Square (Research Square), Feb 14, 2023

We consider general classes of gradient models on regular trees with values in a countable Abelia... more We consider general classes of gradient models on regular trees with values in a countable Abelian group S such as Z or Zq, in regimes of strong coupling (or low temperature). This includes unbounded spin models like the p-SOS model and finite-alphabet clock models. We prove the existence of families of distinct homogeneous tree-indexed Markov chain Gibbs states µA whose single-site marginals concentrate on a given finite subset A ⊂ S of spin values, under a strong coupling condition for the interaction, depending only on the cardinality |A| of A. The existence of such states is a new and robust phenomenon which is of particular relevance for infinite spin models. These states are not convex combinations of each other, and in particular the states with |A| ≥ 2 can not be decomposed into homogeneous Markov-chain Gibbs states with a singlevalued concentration center. As a further application of the method we obtain moreover the existence of new types of Z-valued gradient Gibbs states, whose single-site marginals do not localize, but whose correlation structure depends on the finite set A.

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

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.