Fixation for Two-Dimensional $${\mathcal {U}}$$-Ising and $${\mathcal {U}}$$-Voter Dynamics

Daniel, Blanquicett,

doi:10.1007/s10955-020-02697-8

Cited by 4 publications

(6 citation statements)

References 16 publications

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“…If r > d, on the other hand, then it is not hard to see that p c (Z d n , N d r ) → 1 as n → ∞, since an uninfected copy of {0, 1} d cannot be infected in the process. Similar results have been proved for some other families of automata in two dimensions [16,31,32,34], and weaker bounds are known for a small number of specific three-dimensional update families [13,35].…”

Section: Introductionsupporting

confidence: 77%

Universality for monotone cellular automata

Balister¹,

Bollobás²,

Morris³

et al. 2022

Preprint

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In this paper we study monotone cellular automata in d dimensions. We develop a general method for bounding the growth of the infected set when the initial configuration is chosen randomly, and then use this method to prove a lower bound on the critical probability for percolation that is sharp up to a constant factor in the exponent for every 'critical' model. This is one of three papers that together confirm the Universality Conjecture of Bollobás, Duminil-Copin, Morris and Smith.

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Section: Introductionsupporting

confidence: 77%

Universality for monotone cellular automata

Balister¹,

Bollobás²,

Morris³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…In particular, the the upper bound for the second case in (6) follows. Therefore, it only remains to cover the first case c = a + b when r − c ∈ {3, .…”

Section: Proof Of Theorem 11: Upper Boundsmentioning

confidence: 97%

“…The study of bootstrap processes on graphs was initiated in 1979 by Chalupa, Leath and Reich [11], and is motivated by problems arising from statistical physics, such as the Glauber dynamics of the zero-temperature Ising model, and kinetically constrained spin models of the liquid-glass transition (see, e.g., [6,16,[18][19][20]). The r-neighbour bootstrap process on a locally finite graph G is a monotone cellular automata on the configuration space {0, 1} V (G) , (we call vertices in state 1 "infected"), evolving in discrete time in the following way: 0 becomes 1 when it has at least r neighbours in state 1, and infected vertices remain infected forever.…”

Section: Introductionmentioning

confidence: 99%

Three-dimensional 2-critical bootstrap percolation: The stable sets approach

Daniel¹

2022

Preprint

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Consider a p-random subset A of initially infected vertices in the discrete cube [L] 3 , and assume that the neighbourhood of each vertex consists of the a i nearest neighbours in the ±e i -directions for each i ∈ {1, 2, 3}, where a 1 a 2 a 3 . Suppose we infect any healthy vertex v ∈ [L] 3 already having r infected neighbours, and that infected sites remain infected forever. In this paper we determine log of the critical length for percolation up to a constant factor, for all r ∈ {a 3 + 1, . . . , a 3 + a 2 } with a 3 a 1 + a 2 . We moreover give upper bounds for all remaining cases a 3 < a 1 + a 2 and believe that they are tight up to a constant factor.

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“…In this paper we generalize (5) by showing that the critical length is doubly exponential in p for each r ∈ {c+b+1, . .…”

Section: Introductionmentioning

confidence: 96%

“…The study of bootstrap processes on graphs was initiated in 1979 by Chalupa, Leath and Reich [10], and is motivated by problems arising from statistical physics, such as the Glauber dynamics of the zero-temperature Ising model, and kinetically constrained spin models of the liquid-glass transition (see, e.g., [5,15,[18][19][20]). The r-neighbour bootstrap process on a locally finite graph G is a monotone cellular automata on the configuration space {0, 1} V (G) , (we call vertices in state 1 "infected"), evolving in discrete time in the following way: 0 becomes 1 when it has at least r neighbours in state 1, and infected vertices remain infected forever.…”

Section: Introductionmentioning

confidence: 99%

The $d$-dimensional bootstrap percolation models with threshold at least double exponential

Daniel¹

2022

Preprint

Self Cite

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Consider a p-random subset A of initially infected vertices in the discrete cube [L] d , and assume that the neighbourhood of each vertex consists of the a i nearest neighbours in the ±e i -directions for each i ∈ {1, 2, . . . , d}, where a 1 a 2 . . . a d . Suppose we infect any healthy vertex v ∈ [L] d already having r infected neighbours, and that infected sites remain infected forever. In this paper we determine the (d − 1)-times iterated logarithm of the critical length for percolation up to a constant factor, for all d-tuples (a 1 , . . . , a d ) and all r ∈ {a 2Moreover, we reduce the problem of determining this (coarse) threshold for all d 3 and all r ∈ {a d + 1, . . . , a 1 + a 2 + • • • + a d }, to that of determining the threshold for all d 3 and all r ∈ {a d + 1, . . . , a d−1 + a d }. L c (N a 1 ,...,a d r , p) := min{L ∈ N : P p ([A] = [L] d ) 1/2}.

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Fixation for Two-Dimensional $${\mathcal {U}}$$-Ising and $${\mathcal {U}}$$-Voter Dynamics

Cited by 4 publications

References 16 publications

Universality for monotone cellular automata

Universality for monotone cellular automata

Three-dimensional 2-critical bootstrap percolation: The stable sets approach

The $d$-dimensional bootstrap percolation models with threshold at least double exponential

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