Continuity for the asymptotic shape in the frog model with random initial configurations

Kubota, Naoki

doi:10.1016/j.spa.2020.04.005

Cited by 3 publications

(2 citation statements)

References 17 publications

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“…For various finite graphs the asymptotics of the first moment when all sites are visited is established in [BFHM20]. Continuity of the asymptotic shape of the discrete-time frog model with respect to the measure µ is established in [Kub20]. The variance of the passage times is sublinear [vHCN19], which implies that a central limit theorem does not hold.…”

Section: The Transitivity and Recurrence Properties Of The Frog Model...mentioning

confidence: 99%

Linear and superlinear spread for continuous-time frog model

Bezborodov¹,

Krueger²

2020

Preprint

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We consider a stochastic growth model on Z d . Start with some active particle at the origin and η(x) sleeping particles at x ∈ Z d , where η(x) are independent random variables distributed according to µ. Active particles perform a continuous-time simple random walk while sleeping particles stay put until the first arrival of an active particle to their location.Upon the arrival all sleeping particles at the site activate at once and start moving according to their own simple random walks. The aim of this paper is to give conditions on µ under which the spread of the process is linear or faster than linear. The main technique is comparison with other stochastic growth models.

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Section: The Transitivity and Recurrence Properties Of The Frog Model...mentioning

confidence: 99%

Linear and superlinear spread for continuous-time frog model

Bezborodov¹,

Krueger²

2020

Preprint

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“…Proof. The argument as in [12,Lemma 4.2] tells us that there exists a constant c such that for all large k,…”

Section: Proposition 44 There Exists a Positive Integermentioning

confidence: 99%

Strict comparison for the Lyapunov exponents of the simple random walk in random potentials

Kubota¹

2020

Preprint

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We consider the simple random walk in i.i.d. nonnegative potentials on the d-dimensional cubic lattice Z d (d ≥ 1). In this model, the so-called Lyapunov exponent describes the cost of traveling for the simple random walk in the potential. The Lyapunov exponent depends on the distribution function of the potential, and the aim of this article is to prove that the Lyapunov exponent is strictly monotone in the distribution function of the potential with the order according to strict dominance. In particular, for the one-dimensional annealed situation, we observe that the Lyapunov exponents can coincide even under the strict dominance. Furthermore, the comparison for the Lyapunov exponent also provides that for the rate function of this model.

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Linear and superlinear spread for stochastic combustion growth process

Bezborodov,

Krueger

2024

Ann. Inst. H. Poincaré Probab. Statist.

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Continuity for the asymptotic shape in the frog model with random initial configurations

Cited by 3 publications

References 17 publications

Linear and superlinear spread for continuous-time frog model

Linear and superlinear spread for continuous-time frog model

Strict comparison for the Lyapunov exponents of the simple random walk in random potentials

Linear and superlinear spread for stochastic combustion growth process

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