A uniqueness condition for Gibbs measures, with application to the 2-dimensional Ising antiferromagnet

Berg, J. van den

doi:10.1007/bf02097061

Cited by 74 publications

(91 citation statements)

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“…In the latter case, the effective differential adhesion parameter β * is negative, which means that we have essentially repulsion between unlike cell types. The patterns that are expected in the long run are of checkerboard type [Blöte and Wu, 1990;van den Berg, 1993;Dobrushin, Kolafa and Shlosman, 1985;Georgii and Higuchi, 2000], and can therefore be easily mistaken with unordered patterns. However, for checkerboard-type patterns, the degree of order, measured for instance by the absolute values of the spatial correlations, is high.…”

Section: Model Predictions For Cell Sorting Behaviormentioning

confidence: 99%

The cellular basis of cell sorting kinetics

Voß-Böhme

Deutsch

2010

Journal of Theoretical Biology

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Cell sorting is a dynamical cooperative phenomenon that is fundamental for tissue morphogenesis and tissue homeostasis. According to Steinberg's differential adhesion hypothesis, the structure of sorted cell aggregates is determined by physical characteristics of the respective tissues, the tissue surface tensions. Steinberg postulated that tissue surface tensions result from quantitative differences in intercellular adhesion. Several experiments in cell cultures as well as in developing organisms support this hypothesis.The question of how tissue surface tension might result from differential adhesion was addressed in some theoretical models. These models describe the cellular interdependence structure once the temporal evolution has stabilized. In general, these models are capable of reproducing sorted patterns. However the model dynamics at the cellular scale are defined implicitly and are not welljustified. The precise mechanism describing how differential adhesion generates the observed sorting kinetics at the tissue level is still unclear.It is necessary to formulate the concepts of cell level kinetics explicitly. Only then it is possible to understand the temporal development at the cellular and tissue scales. Here we argue that individual cell mobility is reduced the more the cells stick to their neighbors. We translate this assumption into a precise mathematical model which belongs to the class of stochastic interacting particle systems. Analyzing this model, we are able to predict the emergent sorting behavior at the population level. We describe qualitatively the geometry of cell segregation depending on the intercellular adhesion parameters. Furthermore, we derive a functional relationship between intercellular adhesion and surface tension and highlight the role of cell mobility in the process of sorting. We show that the interaction between the cells and the boundary of a confining vessel has a major impact on the sorting geometry.Keywords: differential adhesion hypothesis, tissue surface tension, phase segregation, stochastic lattice gas, interacting particle system 2000 MSC: 92C15, 92C17, 92C37 * Corresponding author Email addresses: anja.voss-boehme@tu-dresden.de (A. Voß-Böhme), andreas.deutsch@tu-dresden.de (A. Deutsch) Preprint submitted to Elsevier 4th December 2009A c c e p t e d m a n u s c r i p t

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Section: Model Predictions For Cell Sorting Behaviormentioning

confidence: 99%

The cellular basis of cell sorting kinetics

Voß-Böhme

Deutsch

2010

Journal of Theoretical Biology

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“…These arguments were introduced in [2], and further developed in [3] - [5] and [10]. There are no totally new ideas in this paper, so in some sense it is a review-like paper.…”

Section: Introductionmentioning

confidence: 98%

“…There are no totally new ideas in this paper, so in some sense it is a review-like paper. However, although the proof of the main result, Theorem 1, is (at least in spirit) 'almost' in [2] and [5] and in [13], the result itself has not been explicitly observed before. This, plus the fact that this result and its proof method may shed some new light on the classical Heilmann-Lieb results, are the main motivation for this paper.…”

Section: Introductionmentioning

confidence: 98%

On the Absence of Phase Transition in the Monomer-Dimer Model

Berg

1999

Perplexing Problems in Probability

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Suppose we cover the set of vertices of a graph G by non-overlapping monomers (singleton sets) and dimers (pairs of vertices corresponding to an edge). Each way to do this is called a monomer-dimer configuration. If G is finite and λ > 0, we define the monomer-dimer distribution for G (with parameter λ) as the probability distribution which assigns to each monomer-dimer configuration a probability proportional to λ |dimers| , where |dimers| is the number of dimers in that configuration. If the graph is infinite, monomer-dimer distributions can be constructed in the standard way, by taking weak limits.We are particularly interested in the monomer-dimer model on (subgraphs of) the d-dimensional cubic lattice. Heilmann and Lieb (1972) prove absence of phase transition, in terms of smoothness properties of certain thermodynamic functions. They do this by studying the location in the complex plane of the zeros of the partition function.We present a different approach and show, by probabilistic arguments, that boundary effects become negligible as the distance to the boundary goes to ∞. This gives absence of phase transition in a related, but generally not equivalent sense as above. However, the decay of boundary effects appears to occur in such a strong way that, by results on general Gibbs systems of Dobrushin and Shlosman (1987) I have tried to write the paper in such a way that it has something interesting for experts as well as beginners in the field of Gibbs measures and phase transitions. As to the latter group of readers, although the paper is not self-contained and part of it assumes some general knowledge about Gibbs measures, I hope that it gives at least an idea of the kind of problems and methods in this field, and that it is entertaining enough to stimulate further study. To avoid confusion: this paper is not about the pure dimer model (in which monomers are not allowed), which has in some sense a more interesting behaviour (see [14]).

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“…Such coupling arguments are also at the origin of the disagreement percolation approach to prove uniqueness of Gibbs states. 11,12 The proof also gives a simple explanation of the uniqueness.…”

Section: Introductionmentioning

confidence: 90%

On the Uniqueness of Gibbs States in the Pirogov–sinai Theory

Kerimov

2006

Int. J. Mod. Phys. B

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We consider models of classical statistical mechanics satisfying natural stability conditions: a finite spin space, translation-periodic finite potential of finite range, a finite number of ground states meeting Peierls or Gertzik-Pirogov-Sinai condition. The Pirogov-Sinai theory describes the phase diagrams of these models at low temperature regimes. By using the method of doubling and mixing of partition functions we give an alternative elementary proof of the uniqueness of limiting Gibbs states at low temperatures in ground state uniqueness region.

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A uniqueness condition for Gibbs measures, with application to the 2-dimensional Ising antiferromagnet

Cited by 74 publications

References 20 publications

The cellular basis of cell sorting kinetics

The cellular basis of cell sorting kinetics

On the Absence of Phase Transition in the Monomer-Dimer Model

On the Uniqueness of Gibbs States in the Pirogov–sinai Theory

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