An Improved Upper Bound for the Critical Probability of the Frog Model on Homogeneous Trees

Lebensztayn, Élcio; Machado, Fábio Prates; Popov, Serguei

doi:10.1007/s10955-004-2051-8

Cited by 28 publications

(56 citation statements)

References 5 publications

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“…in the definition of G n . This is the reason why the upper bound we establish here coincides with the one conjectured by Lebensztayn et al [12].…”

Section: (D)supporting

confidence: 92%

“…In this paper, we derive a new upper bound for the critical value p c that separates these two regimes for the model on homogeneous trees. This upper bound has been conjectured by Lebensztayn et al [12]. For the proof, we construct an approximating sequence of processes that lead directly to the upper bound.…”

Section: Introductionmentioning

confidence: 54%

“…We underline that φ n equals the function G n obtained through the alternative method described by Lebensztayn et al [12], in the remark at the end of Section 4, for constructing another approximating function G n to F n , such that F n (b) ≥ G n (b) for every b. That is, φ n is identical to the function G n that one obtains when the…”

Section: (D)mentioning

confidence: 91%

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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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“…in the definition of G n . This is the reason why the upper bound we establish here coincides with the one conjectured by Lebensztayn et al [12].…”

Section: (D)supporting

confidence: 92%

Section: Introductionmentioning

confidence: 54%

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

2019

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“…Shape theorems on Z d can be found in [2,3]. Phase transitions for the model where particles have a G(1 − p)-distributed lifespan, are investigated in [2,8,12,16]. Recently, in [11], global survival of an asymmetric inhomogeneous random walk system on Z has been studied (in that model particles die after L steps without activation).…”

Section: Introductionmentioning

confidence: 99%

Local and Global Survival for Nonhomogeneous Random Walk Systems on Z

Bertacchi

Machado²,

Zucca

2014

Adv. Appl. Probab.

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Abstract. We study an interacting random walk system on Z where at time 0 there is an active particle at 0 and one inactive particle on each site n ≥ 1. Particles become active when hit by another active particle. Once activated, the particle starting at n performs an asymmetric, translation invariant, nearest neighbor random walk with left jump probability ln. We give conditions for global survival, local survival and infinite activation both in the case where all particles are immortal and in the case where particles have geometrically distributed lifespan (with parameter depending on the starting location of the particle). More precisely, once activated, the particle at n survives at each step with probability pn ∈ [0, 1]. In particular, in the immortal case, we prove a 0-1 law for the probability of local survival when all particles drift to the right. Besides that, we give sufficient conditions for local survival or local extinction when all particles drift to the left. In the mortal case, we provide sufficient conditions for global survival, local survival and local extinction (which apply to the immortal case with mixed drifts as well). Analysis of explicit examples is provided: we describe completely the phase diagram in the cases 1/2 − ln ∼ ±1/n α , pn = 1 and 1/2 − ln ∼ ±1/n α , 1 − pn ∼ 1/n β (where α, β > 0).

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confidence: 99%

Law of large numbers in an epidemic model

Zhukovskii

2012

Dokl. Math.

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113Over the last decade, a huge number of publica tions have been devoted to the construction of epi demic models on graphs (see, e.g., [1][2][3][4][5][6]) and to the study of probability properties of these models. An additional stimulus was provided by the Internet, since epidemic models play an important role in the study of the evolution of web graphs. Many similar models were described in [7]. This paper deals with a general ization of the model proposed by Machado, Mashu rian, and Matzinger [8], who considered the following situation. Let {x 1 , x 2 , …, x n } be a set of points. At the time t = 1, there is a single particle at each point. The point x 1 contains an active particle, while the other points contain inactive ones. At each time t, exactly one active particle with an index having a uniform dis tribution on the index set of active particles available at t moves to a point x ξ(t) . The random variable ξ(t) also has a uniform distribution on {1, 2, …, n}. If the parti cle moves to a point with an inactive particle, then the latter is activated and the former remains active as well. An active particle dies if it moves to a point where there is or was an active particle (if there is an active particle at this point, it survives). Specifically, if a par ticle moves to its own point, it dies. Let D n (t) denote the number of inactive particles at the time t and σ n be the time at which the last active particle dies. The fol lowing important result was derived for the random variable X n = n -D n (σ n ) in [8].Theorem 1 [8]. Let q be a unique solution of the equation 2p = -ln(1 -p) in the interval (0, 1), and let σ = . Then we have the central limit theoremWe study a more complicated model, assuming that several particles move at each time and the number ofthese particles is random. Consider the sets {x 1 (n),x 2 (n), …, x n (n)}, n ∈ ‫.ގ‬ As in the above model, the time t is discrete and we assume that there is a single particle at each point at the time t = 1. The point x 1 = x 1 (n) contains an active particle, while the other point con tain inactive ones. For each n ∈ ‫,ގ‬ the particles are indexed according to their original order (at the time 1, the ith particle is at the point x i ). At each time, an active particle moves with probability p irrespective of the other active particles. Moreover, for a moving particle, the probability of hitting any point is . If a particle moves to a point where there was (or is) an active parti cle, then the former instantly dies. If a particle moves to a point with an inactive particle, then the latter becomes active and the former survives. If several particles move to the same point with an inactive particle at the time, then all of them survive and the inactive particle is acti vated. Inactive and active particles are called alive. Assume that all the considered random variables are defined on some probability space (Ω, Ᏺ, P). Let A n (t) denote the number of active particles at the time t, while D n (t), as before, denote the number of inactive particles. The...

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An Improved Upper Bound for the Critical Probability of the Frog Model on Homogeneous Trees

Cited by 28 publications

References 5 publications

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

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

Local and Global Survival for Nonhomogeneous Random Walk Systems on Z

Law of large numbers in an epidemic model

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