Frozen percolation on the binary tree is nonendogenous

Ráth, Balázs; Swart, Jan M.; Terpai, Tamás

doi:10.48550/arxiv.1910.09213

Cited by 2 publications

(8 citation statements)

References 18 publications

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“…Frozen percolation and forest fires have also been considered on various other graphs. As mentioned earlier, rather explicit quantitative results can be derived for frozen percolation when the graph G is a tree [1], and this special case was further studied in [35] and [27], in particular. Related processes have also been analyzed on the one-dimensional lattice (see e.g.…”

Section: Other Processesmentioning

confidence: 99%

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Near-critical avalanches in 2D frozen percolation and forest fires

Lam¹,

Nolin²

2021

Preprint

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We study two closely related processes on the triangular lattice: frozen percolation, where connected components of occupied vertices freeze (they stop growing) as soon as they contain at least N vertices, and forest fire processes, where connected components burn (they become entirely vacant) at rate ζ > 0. In this paper, we prove that when the density of occupied sites approaches the critical threshold for Bernoulli percolation, both processes display a striking phenomenon: the appearance of near-critical "avalanches".More specifically, we analyze the avalanches, all the way up to the natural characteristic scale of each model. For frozen percolation, we show in particular that the number of frozen clusters surrounding a given vertex is asymptotically equivalent to (log(96/5)) −1 log log N as N → ∞. A similar mechanism underlies forest fires, enabling us to obtain an analogous result for these processes, but with substantially more work: the number of burnt clusters is equivalent to (log(96/41)) −1 log log(ζ −1 ) as ζ 0. This constitutes an important step toward understanding the self-organized critical behavior of such processes.

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Section: Other Processesmentioning

confidence: 99%

“…[5]), and on the complete graph (see [28,26]). The reader can consult Section 1.7 of [27] for an extensive list of references, containing a brief summary of each paper.…”

Section: Other Processesmentioning

confidence: 99%

Near-critical avalanches in 2D frozen percolation and forest fires

Lam¹,

Nolin²

2021

Preprint

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“…Under natural additional assumptions, such a process is even unique in law. This was partially already observed in [Ald00] and made more precise in [RST19,Thm 2]. The problem of almost sure uniqueness stayed open for 19 years, but has recently been solved negatively in [RST19, Thm 3], where it is shown that the question whether a given edge freezes cannot be decided only by looking at the activation times of all edges.…”

Section: Introductionmentioning

confidence: 95%

“…In the remainder of the present subsection, which can be skipped at a first reading, we explain how our set-up relates to the definition of the Marked Binary Branching Tree (MBBT) introduced in [RST19]. Let…”

Section: Frozen Percolation On the Mbbtmentioning

confidence: 99%

“…It turns out that it is mathematically simpler to formulate our results for frozen percolation on a certain oriented tree, the Marked Binary Branching Tree (MBBT), a random oriented continuum tree introduced in [RST19]. Using methods of Section 3 of that paper, our results can also be translated into results for the unoriented 3-regular tree.…”

Section: Introductionmentioning

confidence: 99%

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A phase transition between endogeny and nonendogeny

Ráth¹,

Swart²,

Márton³

2021

Preprint

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The Marked Binary Branching Tree (MBBT) is the family tree of a rate one binary branching process, on which points have been generated according to a rate one Poisson point process, with i.i.d. uniformly distributed activation times assigned to the points. In frozen percolation on the MBBT, initially, all points are closed, but as time progresses points can become either frozen or open. Points become open at their activation times provided they have not become frozen before. Open points connect the parts of the tree below and above it and one says that a point percolates if the tree above it is infinite. We consider a version of frozen percolation on the MBBT in which at times of the form θ n , all points that percolate are frozen. The limiting model for θ → 1, in which points freeze as soon as they percolate, has been studied before by Ráth, Swart, and Terpai. We extend their results by showing that there exists a 0 < θ * < 1 such that the model is endogenous for θ ≤ θ * but not for θ > θ * . This means that for θ ≤ θ * , frozen percolation is a.s. determined by the MBBT but for θ > θ * one needs additional randomness to describe it.

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Frozen percolation on the binary tree is nonendogenous

Cited by 2 publications

References 18 publications

Near-critical avalanches in 2D frozen percolation and forest fires

Near-critical avalanches in 2D frozen percolation and forest fires

A phase transition between endogeny and nonendogeny

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