Deviation bounds for the first passage time in the frog model

Kubota, Naoki

doi:10.1017/apr.2019.8

Cited by 2 publications

(1 citation statement)

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“…The question of transience and recurrence on trees is explored in [HJJ16b,HJJ16a,JJ16,Ros17a]. Recent work has looked at the waking behavior on finite trees [Her18], and the passage time to a distinguished set of vertices in Z d [Kub16]. The article [HJJ17] establishes linear expansion of the set of activated frogs on regular trees, and [HJJ18] pins down different regimes for rapid and slow cover time for the frog model on finite trees.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic behavior of the Brownian frog model

Beckman¹,

Dinan²,

Durrett³

et al. 2018

Electron. J. Probab.

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E l e c t r o n i c J o u r n a l o f P r o b a b i l i t y Electron. AbstractWe introduce an extension of the frog model to Euclidean space and prove properties for the spread of active particles. Fix r > 0 and place a particle at each point x of a unit intensity Poisson point process P ⊆ R d − B(0, r). Around each point in P, put a ball of radius r. A particle at the origin performs Brownian motion. When it hits the ball around x for some x ∈ P, new particles begin independent Brownian motions from the centers of the balls in the cluster containing x. Subsequent visits to the cluster do nothing. This waking process continues indefinitely. For r smaller than the critical threshold of continuum percolation, we show that the set of activated points in P approximates a linearly expanding ball. Moreover, in any fixed ball the set of active particles converges to a unit intensity Poisson point process. Z d with one particle-per-site and simple random walk paths. Initially it was known as the egg model, but was shortly thereafter relabeled the frog model. The "branching," i.e. activation of sleeping particles, is constrained by the number of points on each site and the paths of all frogs in the process. This makes for a rather nuanced process that exhibits behavior different from a single or a branching random walk. Likewise, the Brownian frog model has features somewhere between Brownian motion and branching Brownian motion.Early papers on the frog model consider the process with one particle-per-site on Z d . The paper [TW99] shows that the root is visited infinitely often in all dimensions.Next, [AMP02a, RS04] proved that the set of visited sites has a limiting shape. The latter works in continuous time, so the proofs and theorem statements are slightly different. The past two decades have further investigated the process by modifying the graph, the number of frogs per site, and the random paths followed by frogs. For example, [AMPR01, Pop01] consider the change in transience/recurrence and the limiting shape with a random configuration of frogs. The articles [GS09, DP14] consider the frog model on the lattice where particles perform a biased random walk. Another modification to the random walk path is where frogs take a geometric number of steps then perish. Various phase transitions are explored in [AMP02b, LMP05]. The question of transience and recurrence on trees is explored in [HJJ16b, HJJ16a, JJ16, Ros17a]. Recent work has looked at the waking behavior on finite trees [Her18], and the passage time to a distinguished set of vertices in Z d [Kub16]. The article [HJJ17] establishes linear expansion of the set of activated frogs on regular trees, and [HJJ18] pins down different regimes for rapid and slow cover time for the frog model on finite trees.A Brownian frog model on R is studied in [Ros17b]. Active particles have a fixed leftward drift, and are placed according to a Poisson point process with intensity f (x). This paper establishes sharp conditions on f that determine whether the model is transient or not...

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