A zero-one law for recurrence and transience of frog processes

Kosygina, Elena; Zerner, Martin

doi:10.1007/s00440-016-0711-7

Cited by 24 publications

(23 citation statements)

References 34 publications

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“…They proved that the frog model on Z d with underlying (irreducible) random walk which has an arbitrary drift to the right is recurrent provided that E[log + (η) d+1 2 ] = ∞. Another sufficient recurrence condition involving the tail behaviour of η is derived in [14]. Kosygina and Zerner proved in [14] a zero-one law under the general condition that the frog trajectories are given by a transitive Markov chain.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Theorem 2.6 ( [14]). For any d ≥ 1 and any nearest neighbour transition function π, we have for FM(d, π) that the probability that the origin is visited infinitely many times by active frogs is either 0 or 1.…”

Section: Some Results About Frogsmentioning

confidence: 99%

“…k that can be changed to 0. As we also know that S 1 k+1 ∈ DS1 k , we have for all k ≤ √ d by (14) and the choice of ε…”

Section: Recurrence For D = 2 and Arbitrary Driftmentioning

confidence: 99%

See 2 more Smart Citations

Recurrence and transience of frogs with drift on $\mathbb{Z} ^d$

Döbler¹,

Gantert²,

Höfelsauer³

et al. 2018

Electron. J. Probab.

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We study the frog model on Z d with drift in dimension d ≥ 2 and establish the existence of transient and recurrent regimes depending on the transition probabilities. We focus on a model in which the particles perform nearest neighbour random walks which are balanced in all but one direction. This gives a model with two parameters. We present conditions on the parameters for recurrence and transience, revealing interesting differences between dimension d = 2 and dimension d ≥ 3. Our proofs make use of (refined) couplings with branching random walks for the transience, and with percolation for the recurrence.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Some Results About Frogsmentioning

confidence: 99%

See 1 more Smart Citation

Recurrence and transience of frogs with drift on $\mathbb{Z} ^d$

Döbler¹,

Gantert²,

Höfelsauer³

et al. 2018

Electron. J. Probab.

View full text Add to dashboard Cite

show abstract

“…The question of recurrence or transience for the frog model on integer lattices and infinite trees is an important topic of current research; see Döbler et al [4], Hoffman et al [8], Kosygina and Zerner [10], Rosenberg [14], and references therein. Other central problems for the model without death are related to the growth of the set of visited vertices, and the movement of the cloud of particles.…”

Section: Introductionmentioning

confidence: 99%

A New Upper Bound for the Critical Probability of the Frog Model on Homogeneous Trees

2019

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We consider the interacting particle system on the homogeneous tree of degree (d + 1), known as frog model. In this model, active particles perform independent random walks, awakening all sleeping particles they encounter, and dying after a random number of jumps, with geometric distribution. We prove an upper bound for the critical parameter of survival of the model, which improves the previously known results. This upper bound was conjectured in a paper by Lebensztayn et al. (J. Stat. Phys., 119(1-2), 331-345, 2005). We also give a closed formula for the upper bound.2010 Mathematics Subject Classification. 60K35, 60J85, 82B26, 82B43.

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“…To finish we prove a 0-1 law for the recursive frog model. This is necessary because the recursive frog model is not covered by the 0-1 law in [KZ17].…”

Section: Introductionmentioning

confidence: 99%

The frog model on trees with drift

Beckman¹,

Frank²,

Jiang³

et al. 2019

Electron. Commun. Probab.

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We provide a uniform upper bound on the minimal drift so that the one-persite frog model on a d-ary tree is recurrent. To do this, we introduce a subprocess that couples across trees with different degrees. Finding couplings for frog models on nested sequences of graphs is known to be difficult. The upper bound comes from combining the coupling with a new, simpler proof that the frog model on a binary tree is recurrent when the drift is sufficiently strong. Additionally, we describe a coupling between frog models on trees for which the degree of the smaller tree divides that of the larger one. This implies that the critical drift has a limit as d tends to infinity along certain subsequences.

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A zero-one law for recurrence and transience of frog processes

Cited by 24 publications

References 34 publications

Recurrence and transience of frogs with drift on $\mathbb{Z} ^d$

Recurrence and transience of frogs with drift on $\mathbb{Z} ^d$

A New Upper Bound for the Critical Probability of the Frog Model on Homogeneous Trees

The frog model on trees with drift

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