Zero-temperature Glauber dynamics on $${\mathbb{Z}^d}$$

Morris, Robert

doi:10.1007/s00440-009-0259-x

Cited by 51 publications

(14 citation statements)

References 30 publications

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“…Such processes (as well as several variations of them) have been used as models to describe several complex phenomena in diverse areas, from jamming transitions and magnetic systems to neuronal activity . Bootstrap percolation processes also have connections with the dynamics of the Ising model at zero temperature , . These processes have also been studied on a variety of graphs, such as trees , grids , lattices on the hyperbolic plane , hypercubes , as well as on several distributions of random graphs .…”

Section: Introductionmentioning

confidence: 99%

A phase transition in the evolution of bootstrap percolation processes on preferential attachment graphs

Abdullah

Fountoulakis

2017

Random Struct Algorithms

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The theme of this paper is the analysis of bootstrap percolation processes on random graphs generated by preferential attachment. This is a class of infection processes where vertices have two states: they are either infected or susceptible. At each round every susceptible vertex which has at least r≥2 infected neighbours becomes infected and remains so forever. Assume that initially a(t) vertices are randomly infected, where t is the total number of vertices of the graph. Suppose also that r

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Section: Introductionmentioning

confidence: 99%

A phase transition in the evolution of bootstrap percolation processes on preferential attachment graphs

Abdullah

Fountoulakis

2017

Random Struct Algorithms

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“…This process is known as r-neighbour bootstrap percolation, and has been extensively studied by mathematicians (see, for example, [3,5,17,27,28,33]), physicists (see [2], and the references therein) and sociologists [25,35], amongst others. It has moreover found applications in the Glauber Dynamics of the Ising model (see [22,31]). …”

Section: Introductionmentioning

confidence: 99%

Graph bootstrap percolation

Balogh

Bollobás

Morris

2012

Random Struct Algorithms

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Abstract. Graph bootstrap percolation is a deterministic cellular automaton which was introduced by Bollobás in 1968, and is defined as follows. Given a graph H, and a set G ⊂ E(K n ) of initially 'infected' edges, we infect, at each time step, a new edge e if there is a copy of H in K n such that e is the only not-yet infected edge of H. We say that G percolates in the H-bootstrap process if eventually every edge of K n is infected. The extremal questions for this model, when H is the complete graph K r , were solved (independently) by Alon, Kalai and Frankl almost thirty years ago. In this paper we study the random questions, and determine the critical probability p c (n, K r ) for the K r -process up to a poly-logarithmic factor. In the case r = 4 we prove a stronger result, and determine the threshold for p c (n, K 4 ).

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“…(As a consequence of Schonmann's proof in [33], we have p c ([n] d , d) = o(1) if d log * n.) Morris [28] also used the techniques of [5] to prove that, in zero-temperature Glauber dynamics on Z d , the critical threshold for fixation converges to 1/2 as d → ∞ (see also [22]). …”

Section: Introductionmentioning

confidence: 99%

Bootstrap Percolation in High Dimensions

Balogh¹,

Bollobás²,

Morris³

2010

Combinator. Probab. Comp.

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Abstract. In r-neighbour bootstrap percolation on a graph G, a set of initially infected vertices A ⊂ V (G) is chosen independently at random, with density p, and new vertices are subsequently infected if they have at least r infected neighbours. The set A is said to percolate if eventually all vertices are infected. Our aim is to understand this process on the grid, [n] d , for arbitrary functions n = n(t), d = d(t) and r = r(t), as t → ∞. The main question is to determine the critical probability p c ([n] d , r) at which percolation becomes likely, and to give bounds on the size of the critical window. In this paper we study this problem when r = 2, for all functions n and d satisfying d log n. The bootstrap process has been extensively studied on [n] d when d is a fixed constant and 2 r d, and in these cases p c ([n] d , r) has recently been determined up to a factor of 1 + o(1) as n → ∞. At the other end of the scale, Balogh and Bollobás determined p c ([2] d , 2) up to a constant factor, and Balogh, Bollobás and Morris determined2+ε , and gave much sharper bounds for the hypercube. Here we prove the following result: let λ be the smallest positive root of the equation

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Zero-temperature Glauber dynamics on $${\mathbb{Z}^d}$$

Cited by 51 publications

References 30 publications

A phase transition in the evolution of bootstrap percolation processes on preferential attachment graphs

A phase transition in the evolution of bootstrap percolation processes on preferential attachment graphs

Graph bootstrap percolation

Bootstrap Percolation in High Dimensions

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