Higher order corrections for anisotropic bootstrap percolation

Duminil-Copin, Hugo; Enter, Aernout van; Hulshof, Tim

doi:10.1007/s00440-017-0808-7

Cited by 16 publications

(22 citation statements)

References 37 publications

(125 reference statements)

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“…A number of variations of the bootstrap process described above have been considered. Holroyd [21,22] [8,13,14,15,26]. Similar but weaker results about the threshold behaviour of a very general class of update rules on Z 2 were proved in [3,7,9].…”

Section: Introductionmentioning

confidence: 63%

An Improved Upper Bound for Bootstrap Percolation in All Dimensions

Uzzell

2019

Combinator. Probab. Comp.

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In r-neighbor bootstrap percolation on the vertex set of a graph G, a set A of initially infected vertices spreads by infecting, at each time step, all uninfected vertices with at least r previously infected neighbors. When the elements of A are chosen independently with some probability p, it is natural to study the critical probability pc(G, r) at which it becomes likely that all of V (G) will eventually become infected. Improving a result of Balogh, Bollobás, and Morris, we give a bound on the second term in the expansion of the critical probability when G = [n] d and d ≥ r ≥ 2. We show that for all d ≥ r ≥ 2 there exists a constant c d,r > 0 such that if n is sufficiently large, thenwhere λ(d, r) is an exact constant and log (k) (n) denotes the k-times iterated natural logarithm of n.1 It follows that there exists α > 0 such that the right-hand side of (A.6) is at least α(log n) 1/2 − d log n, which is what we wanted.

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Section: Introductionmentioning

confidence: 63%

An Improved Upper Bound for Bootstrap Percolation in All Dimensions

Uzzell

2019

Combinator. Probab. Comp.

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“…We remark that an example of an update family satisfying the conditions of Conjecture 4 is the so-called anisotropic model (see, e.g., [18,19]) whose update family consists of all subsets of size 3 of the set (−2, 0), (−1, 0), (1, 0), (2, 0), (0, 1), (0, −1) .…”

Section: Conjecturementioning

confidence: 99%

Universality Results for Kinetically Constrained Spin Models in Two Dimensions

Martinelli

Morris

Toninelli

2018

Commun. Math. Phys.

105

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Kinetically constrained models (KCM) are reversible interacting particle systems on Z d with continuous time Markov dynamics of Glauber type, which represent a natural stochastic (and non-monotone) counterpart of the family of cellular automata known as U-bootstrap percolation. KCM also display some of the peculiar features of the so-called "glassy dynamics", and as such they are extensively used in the physics literature to model the liquid-glass transition, a major and longstanding open problem in condensed matter physics.We consider two-dimensional KCM with update rule U, and focus on proving universality results for the mean infection time of the origin, in the same spirit as those recently established in the setting of U-bootstrap percolation. We first identify what we believe are the correct universality classes, which turn out to be different from those of U-bootstrap percolation. We then prove universal upper bounds on the mean infection time within each class, which we conjecture to be sharp up to logarithmic corrections. In certain cases, including all supercritical models, and the well-known Duarte model, our conjecture has recently been confirmed in [28]. In fact, in these cases our upper bound is sharp up to a constant factor in the exponent. For certain classes of update rules, it turns out that the infection time of the KCM diverges much faster than for the corresponding U-bootstrap process when the equilibrium density of infected sites goes to zero. This is due to the occurrence of energy barriers which determine the dominant behaviour for KCM, but which do not matter for the monotone bootstrap dynamics.

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“…Such sharp or sharper bounds have been obtained for a handful of other specific models [4,8,9], but still remain open in general. However, the level of precision of the Aizenman-Lebowitz result was established in full generality for critical models by Bollobás, Duminil-Copin, Morris and Smith [5].…”

Section: Introductionmentioning

confidence: 97%

Complexity of Two-dimensional Bootstrap Percolation Difficulty: Algorithm and NP-Hardness

Hartarsky¹,

Mezei²

2020

SIAM J. Discrete Math.

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Bootstrap percolation is a class of cellular automata with random initial state. Two-dimensional bootstrap percolation models have three rough universality classes, the most studied being the "critical" one. For this class the scaling of the quantity of greatest interest (the critical probability) was determined by Bollobás, Duminil-Copin, Morris and Smith [5] in terms of a simply defined combinatorial quantity called "difficulty", so the subject seemed closed up to finding sharper results. However, the computation of the difficulty was never considered. In this paper we provide the first algorithm to determine this quantity, which is, surprisingly, not as easy as the definition leads to thinking. The proof also provides some explicit upper bounds, which are of use for bootstrap percolation. On the other hand, we also prove the negative result that computing the difficulty of a critical model is NP-hard. This two-dimensional picture contrasts with an upcoming result of Balister, Bollobás, Morris and Smith [3] on uncomputability in higher dimensions. The proof of NP-hardness is achieved by a technical reduction to the Set Cover problem.

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Higher order corrections for anisotropic bootstrap percolation

Cited by 16 publications

References 37 publications

An Improved Upper Bound for Bootstrap Percolation in All Dimensions

An Improved Upper Bound for Bootstrap Percolation in All Dimensions

Universality Results for Kinetically Constrained Spin Models in Two Dimensions

Complexity of Two-dimensional Bootstrap Percolation Difficulty: Algorithm and NP-Hardness

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