arXiv subjects

Christof Külske

Publications and source records attributed to Christof Külske.

11 recordsLinked to original sources

arXiv2026

$d$-dimensional spherical ferromagnets in random fields: Metastates, continuous symmetry breaking, and spin-glass features

We study the large-volume behavior of the spherical model for $d$-dimensional local spins, in the presence of $d$-dimensional random fields, for $d\geq 2$. We compare two models, one with volume-scaled random fields, and another one with non-scaled random fields, on the level of Aizenman-Wehr metastates, Newman-Stein metastates, as well as overlap distributions. We show that in $d\geq 2$ the metastates are fully supported on a continuity of random product states, with weights which we describe, for both models. For the non-scaled random fields, the set of a.s. cluster points of Gibbs measures contains these product states, but behaves differently in the 'recurrent' spin dimension $d=2$ where it also contains non-trivial mixtures of tilted measures. For the scaled model, moreover the overlap distribution displays spin-glass characteristics, as it is non-self averaging, and shows replica symmetry breaking, although it is ultrametric if and only if $d=1$. For $d\geq 2$ it oscillates chaotically on a set of continuous distributions for large volumes, while the limiting set contains only discrete distributions in $d=1$. Our results are based on concentration estimates, analysis of Gibbs measures in finite but large volumes, and the asymptotics of $d$-dimensional random walks and their spherical projections.

math-ph↗

arXiv2026

Dynamical Gibbs-non-Gibbs transitions for finite-alphabet models on trees

We study finite-alphabet spin models on trees evolving under independent symmetric spin-flip dynamics. On the lattice the time-evolved plus measure of the low temperature Ising model in zero external field was shown to be non-quasilocal for all sufficiently large times in \cite{EnFeHoRe02}. On the tree however the time-evolved plus phase of the Ising model behaves differently, as the quasilocal Gibbs property is first lost, but then recovered at a later time \cite{EnErIaKu12}. We first extend this result by showing that large-time reentry into the quasilocal Gibbs regime on the tree happens more generally for all finite alphabet models, when the dynamics is started in $f$-stable Gibbs states. These are defined in terms of a time-independent sharp condition on the homogeneous recursion $f$ describing the model, which can be checked explicitly. As our second and opposite result, we develop a method for proving large-time persistence of non-quasilocality of time-evolved measures. We show that even all configurations can become bad at large times, when the initial Gibbs measure corresponds to an $f$-saddle and give an explicit illustration for the Potts model. In our proofs we study the influence of spatially inhomogeneous perturbations to the solutions of a time-dependent fixed point problem.

math.PR↗

arXiv2026

Lattice random-field Widom--Rowlinson models

We consider the Widom--Rowlinson model on $\mathbb{Z}^d$ subject to a symmetric i.i.d.\ random field. We prove that for dimensions $d\le 2$ any non-trivial random field leads to an absence of a phase transition. In contrast, in dimensions $d\ge 3$ and for Gaussian random fields, phase-transition behavior of the model is maintained for sufficiently large densities of occupied sites. This extends the general picture known from the classical random-field Ising model to the random-field Widom--Rowlinson model. Following the general proof route of Aizenman--Wehr as well as Ding--Zhuang, our main contribution rests on adequate notions of contours and their associated generalized spin-flip operation to deal with hard-core repulsions.

math.PR↗

arXiv2026

Locality properties for discrete and continuum Widom--Rowlinson models in random environments

We consider the Widom--Rowlinson model in which hard balls of two possible colors are constrained to a hard-core repulsion between particles of different colors, in quenched random environments. These random environments model spatially dependent preferences for the attachment of balls. We investigate the possibility to represent the joint process of environment and infinite-volume Widom--Rowlinson measure in terms of continuous (quasilocal) Papangelou intensities. We show that this is not always possible: In the case of the symmetric Widom-Rowlinson model on a non-percolating environment, we can explicitly construct a discontinuity coming from the environment. This is a new phenomenon for systems of continuous particles, but it can be understood as a continuous-space echo of a simpler non-locality phenomenon known to appear for the diluted Ising model (Griffiths singularity random field) on the lattice, as we explain in the course of the proof.

math.PR↗

arXiv2024

Gibbs Properties of the Bernoulli field on inhomogeneous trees under the removal of isolated sites

We consider the i.i.d. Bernoulli field $μ_p$ with occupation density $p \in (0,1)$ on a possibly non-regular countably infinite tree with bounded degrees. For large $p$, we show that the quasilocal Gibbs property, i.e. compatibility with a suitable quasilocal specification, is lost under the deterministic transformation which removes all isolated ones and replaces them by zeros, while a quasilocal specification does exist at small $p$. Our results provide an example for an independent field in a spatially non-homogeneous setup which loses the quasilocal Gibbs property under a local deterministic transformation.

math.PR↗

arXiv2018

Extremal decomposition for random Gibbs measures: From general metastates to metastates on extremal random Gibbs measures

The concept of metastate measures on the states of a random spin system was introduced to be able to treat the large-volume asymptotics for complex quenched random systems, like spin glasses, which may exhibit chaotic volume dependence in the strong-coupling regime. We consider the general issue of the extremal decomposition for Gibbsian specifications which depend measurably on a parameter that may describe a whole random environment in the infinite volume. Given a random Gibbs measure, as a measurable map from the environment space, we prove measurability of its decomposition measure on pure states at fixed environment, with respect to the environment. As a general corollary we obtain that, for any metastate, there is an associated decomposition metastate, which is supported on the extremes for almost all environments, and which has the same barycenter.

math.PR↗

arXiv2018

Gibbsian representation for point processes via hyperedge potentials

We consider marked point processes on the d-dimensional euclidean space, defined in terms of a quasilocal specification based on marked Poisson point processes. We investigate the possibility of constructing absolutely-summable Hamiltonians in terms of hyperedge potentials in the sense of Georgii et al. These potentials are a natural generalization of physical multi-body potentials which are useful in models of stochastic geometry. We prove that such representations can be achieved, under appropriate locality conditions of the specification. As an illustration we also provide such potential representations for the Widom-Rowlinson model under independent spin-flip time-evolution. Our paper draws a link between the abstract theory of point processes in infinite volume, the study of measures under transformations, and statistical mechanics of systems of point particles.

math.PR↗

arXiv2014

Uniqueness of gradient Gibbs measures with disorder

We consider - in uniformly strictly convex potential regime - two versions of random gradient models with disorder. In model (A) the interface feels a bulk term of random fields while in model (B) the disorder enters though the potential acting on the gradients. We assume a general distribution on the disorder with uniformly-bounded finite second moments. It is well known that for gradient models without disorder there are no Gibbs measures in infinite-volume in dimension $d = 2$, while there are shift-invariant gradient Gibbs measures describing an infinite-volume distribution for the gradients of the field, as was shown by Funaki and Spohn. Van Enter and Kuelske proved in 2008 that adding a disorder term as in model (A) prohibits the existence of such gradient Gibbs measures for general interaction potentials in $d = 2$. In Cotar and Kuelske (2012) we proved the existence of shift-covariant random gradient Gibbs measures for model (A) when $d\geq 3$, the disorder is i.i.d and has mean zero, and for model (B) when $d\geq 1$ and the disorder has stationary distribution. In the present paper, we prove existence and uniqueness of shift-covariant random gradient Gibbs measures with a given expected tilt $u\in R^d$ and with the corresponding annealed measure being ergodic: for model (A) when $d\geq 3$ and the disordered random fields are i.i.d. and symmetrically-distributed, and for model (B) when $d\geq 1$ and for any stationary disorder dependence structure. We also compute for both models for any gradient Gibbs measure constructed as in Cotar and Kuelske (2012), when the disorder is i.i.d. and its distribution satisfies a Poincaré inequality assumption, the optimal decay of covariances with respect to the averaged-over-the-disorder gradient Gibbs measure.

math.PR↗

arXiv2014

A class of nonergodic interacting particle systems with unique invariant measure

We consider a class of discrete $q$-state spin models defined in terms of a translation-invariant quasilocal specification with discrete clock-rotation invariance which have extremal Gibbs measures $μ'_φ$ labeled by the uncountably many values of $φ$ in the one-dimensional sphere (introduced by van Enter, Opoku, Külske [J. Phys. A 44 (2011) 475002, 11]). In the present paper we construct an associated Markov jump process with quasilocal rates whose semigroup $(S_t)_{t\geq0}$ acts by a continuous rotation $S_t(μ'_φ)=μ'_{φ+t}$. As a consequence our construction provides examples of interacting particle systems with unique translation-invariant invariant measure, which is not long-time limit of all starting measures, answering an old question (compare Liggett [Interacting Particle Systems (1985) Springer], question four, Chapter one). The construction of this particle system is inspired by recent conjectures of Maes and Shlosman about the intermediate temperature regime of the nearest-neighbor clock model. We define our generator of the interacting particle system as a (noncommuting) sum of the rotation part and a Glauber part. Technically the paper rests on the control of the spread of weak nonlocalities and relative entropy-methods, both in equilibrium and dynamically, based on Dobrushin-uniqueness bounds for conditional measures.

math.PR↗

arXiv2012

Existence of random gradient states

We consider two versions of random gradient models. In model A the interface feels a bulk term of random fields while in model B the disorder enters through the potential acting on the gradients. It is well known that for gradient models without disorder there are no Gibbs measures in infinite-volume in dimension d=2, while there are "gradient Gibbs measures" describing an infinite-volume distribution for the gradients of the field, as was shown by Funaki and Spohn. Van Enter and Külske proved that adding a disorder term as in model A prohibits the existence of such gradient Gibbs measures for general interaction potentials in $d=2$. In the present paper we prove the existence of shift-covariant gradient Gibbs measures with a given tilt $u\in \mathbb{R}^d$ for model A when $d\geq3$ and the disorder has mean zero, and for model B when $d\geq1$. When the disorder has nonzero mean in model A, there are no shift-covariant gradient Gibbs measures for $d\ge3$. We also prove similar results of existence/nonexistence of the surface tension for the two models and give the characteristic properties of the respective surface tensions.

math.PR↗

arXiv2008

Nonexistence of random gradient Gibbs measures in continuous interface models in $d=2$

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-volume distribution for the increments of the field, as was shown by Funaki and Spohn. In the present paper we show that adding a disorder term prohibits the existence of such gradient Gibbs measures for general interaction potentials in $d=2$. This nonexistence result generalizes the simple case of Gaussian fields where it follows from an explicit computation. In $d=3$ where random gradient Gibbs measures are expected to exist, our method provides a lower bound of the order of the inverse of the distance on the decay of correlations of Gibbs expectations w.r.t. the distribution of the random environment.

math.PR↗