Phase Transition for the Frog Model

Alves, O. S. M.; Machado, Fábio Prates; Popov, Serguei

doi:10.1214/ejp.v7-115

Cited by 60 publications

(77 citation statements)

References 7 publications

(5 reference statements)

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Moreover, if D A = D B then there exists a constant C 2 > 0 such that for each constant K > 0 and for sufficiently large t [17]. These two theorems suggests that a "shape theorem" may hold: t −1 B(t) converges to a non-random set B 0 which implies that the growth rate of B(t) is linear in t. Note that when D A = 0 which implies that susceptible particles do not move, the model degenerates to what is known as the frog model [3]. In this case there exists a full shape theorem, as follows: ∃ a non random set B 0 such that for all 0 < ε < 1…”

Section: Epidemics On Agentsmentioning

confidence: 99%

The End Time of SIS Epidemics Driven by Random Walks on Edge-Transitive Graphs

2020

View full text Add to dashboard Cite

Network epidemics is a ubiquitous model that can represent different phenomena and finds applications in various domains. Among its various characteristics, a fundamental question concerns the time when an epidemic stops propagating. We investigate this characteristic on a SIS epidemic induced by agents that move according to independent continuous time random walks on a finite graph: Agents can either be infected (I) or susceptible (S), and infection occurs when two agents with different epidemic states meet in a node. After a random recovery time, an infected agent returns to state S and can be infected again. The End of Epidemic (EoE) denotes the first time where all agents are in state S, since after this moment no further infections can occur and the epidemic stops. For the case of two agents on edge-transitive graphs, we characterize EoE as a function of the network structure by relating the Laplace transform of EoE to the Laplace transform of the meeting time of two random walks. Interestingly, this analysis shows a separation between the effect of network structure and epidemic dynamics. We then study the asymptotic behavior of EoE (asymptotically in the size of the graph) under different parameter scalings, identifying regimes where EoE converges in distribution to a proper random variable or to infinity. We also highlight the impact of different graph structures on EoE, characterizing it under complete graphs, complete bipartite graphs, and rings.

show abstract

Section: Epidemics On Agentsmentioning

confidence: 99%

The End Time of SIS Epidemics Driven by Random Walks on Edge-Transitive Graphs

2020

View full text Add to dashboard Cite

show abstract

“…Local survival is nontrivial unless l n = 1/2 for some n ∈ N. In order to understand what the difficulties one encounters are, think of the case where all particles drift to the right (we refer to this situation as the right drift case): infinite activation is guaranteed but local survival is not. Theorem 2.1 (1) states that, in this case, the probability of local survival obeys a 0-1 law. Roughly speaking (see Corollary 2.2) in the right drift case, if l n ↑ 1 2 sufficiently fast, then we have almost sure local survival, otherwise we have local extinction.…”

Section: Introductionmentioning

confidence: 96%

Local and Global Survival for Nonhomogeneous Random Walk Systems on Z

Bertacchi

Machado²,

Zucca

2014

Adv. Appl. Probab.

View full text Add to dashboard Cite

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).

show abstract

“…Whenever an active particle hits a vertex containing a sleeping one, the latter is activated, and starts its own life and trajectory on G, in an independent manner. We denote by FM(G, p) the frog model on G, with survival parameter p, referring the reader to Alves et al [1] for the formal definition of the model.…”

Section: Introductionmentioning

confidence: 99%

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

2019

View full text Add to dashboard Cite

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.

show abstract

Phase Transition for the Frog Model

Cited by 60 publications

References 7 publications

The End Time of SIS Epidemics Driven by Random Walks on Edge-Transitive Graphs

The End Time of SIS Epidemics Driven by Random Walks on Edge-Transitive Graphs

Local and Global Survival for Nonhomogeneous Random Walk Systems on Z

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

Contact Info

Product

Resources

About