Recurrence of the frog model
on the 3,2-alternating tree

Rosenberg, Josh

doi:10.30757/alea.v15-30

Cited by 13 publications

(10 citation statements)

References 4 publications

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“…For the next step in the proof of the claim, we observe that because the harmonic measure of any single level 1 vertex is bounded above by 1 − β for some β > 0 (see explanation for this at the beginning of the proof of Lemma 4.4), this implies that it follows from Claim 1, along with Markov's inequality, that the expectation of the larger expression in (12) with respect to AGW n is bounded above by…”

Section: Proof Of Recurrencementioning

confidence: 94%

“…Since, by (12), this value must be greater than or equal to the expression to the left of the inequality in (10), it follows that (10) must hold for…”

Section: Proof Of Recurrencementioning

confidence: 99%

“…While there have been efforts to break free of this constraint, they have been limited in scope. One such attempt was made by the second author in [12], where he established recurrence for the one per-site model on the 3, 2-alternating tree. Yet even in this case, the proof exploits the self-similarity of the bi-regular tree, specifically relying on the fact that (as with all of the other graphs referenced above) the 3, 2-alternating tree is quasi-transitive, meaning that the set of vertices can be partitioned into finitely many sets so that for each pair of vertices in the same set there exists a graph automorphism mapping one to the other.…”

Section: Introductionmentioning

confidence: 99%

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The frog model on Galton-Watson trees

Michelen,

Rosenberg

2019

Preprint

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We consider an interacting particle system on trees known as the frog model: initially, a single active particle begins at the root and i.i.d. Poiss(λ) many inactive particles are placed at each non-root vertex. Active particles perform discrete time simple random walk and activate the inactive particles they encounter. We show that for Galton-Watson trees with offspring distributions Z satisfying P(Z ≥ 2) = 1 and E[Z 4+ǫ ] < ∞ for some ǫ > 0, there is a critical value λc ∈ (0, ∞) separating recurrent and transient regimes for almost surely every tree, thereby answering a question of Hoffman-Johnson-Junge. In addition, we also establish that this critical parameter depends on the entire offspring distribution, not just the maximum value of Z, answering another question of Hoffman-Johnson-Junge and showing that the frog model and contact process behave differently on Galton-Watson trees.

show abstract

Section: Proof Of Recurrencementioning

confidence: 94%

“…Since, by (12), this value must be greater than or equal to the expression to the left of the inequality in (10), it follows that (10) must hold for…”

Section: Proof Of Recurrencementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The frog model on Galton-Watson trees

Michelen,

Rosenberg

2019

Preprint

Self Cite

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“…Hoffman et al [18] establish that the frog model is recurrent on the binary tree, but it is transient on d-ary trees, for d ≥ 5. We also refer to Hoffman et al [17], Johnson and Junge [23] and Rosenberg [40]. The subjects in (ii) are investigated mainly for processes on Z d (see, for instance, Alves et al [2], Ramírez and Sidoravicius [38], Höfelsauer and Weidner [16]), and lately analyzed on d-ary trees (Hoffman et al [20]).…”

Section: Introductionmentioning

confidence: 99%

Laws of large numbers for the frog model on the complete graph

Lebensztayn

Estrada

2019

Journal of Mathematical Physics

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The frog model is a stochastic model for the spreading of an epidemic on a graph, in which a dormant particle starts to perform a simple random walk on the graph and to awake other particles, once it becomes active. We study two versions of the frog model on the complete graph with N + 1 vertices. In the first version we consider, active particles have geometrically distributed lifetimes. In the second version, the displacement of each awakened particle lasts until it hits a vertex already visited by the process. For each model, we prove that as N → ∞, the trajectory of the process is well approximated by a three-dimensional discrete-time dynamical system. We also study the long-term behavior of the corresponding deterministic systems.2010 Mathematics Subject Classification. 60K35, 60J10, 92B05.

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“…For the frog model, most of the work has involved studying the process on Z d , T d and recently on d-ary trees. As far as we know, only Rosenberg [15] considers another kind of tree, proving the recurrence of the process (without death) in a 3, 2-alternating tree (in which the generations of vertices alternate between having 2 and 3 children).…”

Section: Introductionmentioning

confidence: 99%

Phase transition for the frog model on biregular trees

Lebensztayn,

Utria

2018

Preprint

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We study the frog model with death on the biregular tree T d1,d2 . Initially, there is a random number of awake and sleeping particles located on the vertices of the tree. Each awake particle moves as a discrete-time independent simple random walk on T d1,d2 and has a probability of death (1 − p) before each step. When an awake particle visits a vertex which has not been visited previously, the sleeping particles placed there are awakened. We prove that this model undergoes a phase transition: for values of p below a critical probability p c , the system dies out almost surely, and for p > p c , the system survives with positive probability. We establish explicit bounds for p c in the case of random initial configuration. For the model starting with one particle per vertex, the critical probability satisfies p c (T d1,d2 ) = 1/2 + Θ(1/d 1 + 1/d 2 ) as d 1 , d 2 → ∞.

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Recurrence of the frog model on the 3,2-alternating tree

Cited by 13 publications

References 4 publications

The frog model on Galton-Watson trees

The frog model on Galton-Watson trees

Laws of large numbers for the frog model on the complete graph

Phase transition for the frog model on biregular trees

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