Frozen percolation on the binary tree is nonendogenous

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

doi:10.1214/21-aop1507

Cited by 2 publications

(5 citation statements)

References 39 publications

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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%

“…(2) Ξ θ the diagonal solution of the bivariate RDE since it is concentrated on {(y, y) : y ∈ I} (see (1.24)). In the special case θ = 1, Theorem 1.14 has been proved in [RST19,Thm 12], where it is moreover shown that the bivariate RDE (1.23) has precisely two solutions in the space M…”

Section: We Call ρmentioning

confidence: 99%

“…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%

“…copies of Y ∅ , and ω is an independent uniformly distributed random variable on [0, 1] × {1, 2}. Proposition 37 of [RST19] classifies all solutions of the RDE (1.17). Expanding on that result, we can prove the following lemma, which is the basis of our proof of Theorem 1.1.…”

Section: Burning Timesmentioning

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

Ráth

Swart

Márton

2022

Electron. J. Probab.

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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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Section: Introductionmentioning

confidence: 95%

Section: We Call ρmentioning

confidence: 99%

Section: Frozen Percolation On the Mbbtmentioning

confidence: 99%

Section: Burning Timesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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