Percolation and number of phases in the two-dimensional Ising model

Georgii, Hans-Otto; Higuchi, Yasunari

doi:10.1063/1.533182

Cited by 23 publications

(28 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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“…Lucio proved this by considering the existence (or not) of infinite +/− clusters on Z 2 and its matching lattice. The full conclusion, without an assumption of partial translation-invariance, was obtained later in independent work of Aizenman, [1], and Higuchi, [32] (see also [22]). Therefore, in two dimensions (unlike three dimensions) there exists no non-translation-invariant Gibbs measure.…”

Section: Ising Modelmentioning

confidence: 75%

The work of Lucio Russo on percolation

Grimmett

2016

Math. Mech. Compl. Sys.

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The contributions of Lucio Russo to the mathematics of percolation and disordered systems are outlined. The context of his work is explained, and its ongoing impact on current work is described and amplified. A personal appreciationPrior to his mid-career move to the history of science in the early 1990s, Lucio Russo enjoyed a very successful and influential career in the theory of probability and disordered systems, in particular of percolation and the Ising model. His ideas have shaped these significant fields of science, and his name will always be associated with a number of fundamental techniques of enduring importance.The author of this memoir is proud to have known Lucio in those days, and to have profited from his work, ideas, and company. He hopes that this brief account of some of Lucio's results will stand as testament to the beauty and impact of his ideas. Scientific summaryLucio Russo has worked principally on the mathematics of percolation, that is, of the existence (or not) of infinite connected clusters within a disordered spatial network. The principal model in this field is the so called percolation model, introduced to mathematicians by Broadbent and Hammersley in 1957, [14]. Consider, for definiteness, the hypercubic lattice ‫ޚ‬ d with d ≥ 2, and let p ∈ [0, 1]. We declare each edge to be open with probability p and closed otherwise, and different edges receive independent states. The main questions are centred around the existence Let C be the open cluster of ‫ޚ‬ d containing the origin. Two functions that play important roles in the theory are the percolation probability θ and the mean cluster size χ given by Lucio contributed a number of fundamental techniques to percolation theory during the period 1978-1988, and the main purpose of the current paper is to describe these and to explore their significance. We mention Russo's formula, the Russo-Seymour-Welsh (RSW) inequalities, his study of percolation surfaces in three dimensions, and of the uniqueness of the infinite open cluster, and finally Russo's approximate zero-one law. Russo's formula and RSW theory have proved of especially lasting value in, for example, recent developments concerning conformal invariance for critical percolation.In Section 8, we mention some of Lucio's results concerning percolation of +/− spins in the two-dimensional Ising model. It was quite a novelty in the 1970s to use percolation as a tool to understand long-range order in the Ising model. Indeed, Lucio's work on the percolation model was motivated in part by his search for rigorous results in statistical mechanics. His approach to the Ising model has been valuable in two dimensions. In more general situations, the correct geometrical model has been recognised since to be the random-cluster model of Fortuin and Kasteleyn (see [26]).This short account is confined to Lucio's contributions to percolation, and does not touch on his work lying closer to ergodic theory and dynamical systems, namely [R2; R6; R8; R11], and neither does it refer to the paper [R7]. A ...

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“…By the random-cluster representation or otherwise, the two-point correlation functions π ± βc (σ x σ y ) are bounded away from 0 for all pairs x, y of vertices. By the main result of [1,16] (see also [10]) and the symmetry of π 0 βc , we have that π 0 βc = 1 2 π + βc + 1 2 π − βc , whence π 0 βc (σ x σ y ) is bounded away from 0. By [12, Thm 5.17], this contradicts the above remark that the percolation-probability of the free-boundary condition random-cluster measure is 0 at β = β sd = β c .…”

Section: Random Even Subgraphs Of Planar Latticesmentioning

confidence: 83%

Random Even Graphs

Grimmett¹,

Janson²

2009

Electron. J. Combin.

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We study a random even subgraph of a finite graph G with a general edge-weight p ∈ (0, 1). We demonstrate how it may be obtained from a certain random-cluster measure on G, and we propose a sampling algorithm based on coupling from the past. A random even subgraph of a planar lattice undergoes a phase transition at the parameter-value 1 2 p c , where p c is the critical point of the q = 2 random-cluster model on the dual lattice. The properties of such a graph are discussed, and are related to Schramm-Löwner evolutions (SLE).

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Percolation and number of phases in the two-dimensional Ising model

Cited by 23 publications

References 9 publications

The cellular basis of cell sorting kinetics

The cellular basis of cell sorting kinetics

The work of Lucio Russo on percolation

Random Even Graphs

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