Phase Transition for the Frog Model on Biregular Trees

Author(s)

Lebensztayn, Elcio ^[1] ; Utria, Jaime ^[2]

Affiliation

^[1] Univ Estadual Campinas, UNICAMP, Inst Math Stat & Sci Computat, Rua Sergio Buarque da Holanda 651, BR-13083859 Campinas, SP - Brazil

^[2] Fluminense Fed Univ UFF, Inst Math & Stat, Rua Prof Marcos Waldemar de Freitas Reis S-N, BR-24210201 Niteroi, RJ - Brazil

• Document type: Journal article
• Source: Markov Processes and Related Fields; v. 26, n. 3, p. 447-466, 2020.
• Web of Science Citations: 0

Abstract

We study the frog model with death on the biregular tree T-d1,T-d2. Initially, there is a random number of active and inactive particles located on the vertices of the tree. Each active particle moves as a discrete-time independent simple random walk on T-d1,T-d2 and has a probability of death (1 - p) before each step. When an active particle visits a vertex which has not been visited previously, the inactive particles placed there are activated. We prove that this model undergoes a phase transition: for values of p below a critical probability p(c), the system dies out almost surely, and for p > p(c), the system survives with positive probability. We establish explicit bounds for p(c) in the case of random initial configuration. For the model starting with one particle per vertex, the critical probability satisfies p(c)(T-d1,T-d2) = 1/2 + Theta(1/d(1) + 1/d(2)) as d(1), d(2) -> infinity. (AU)

• FAPESP's process: 17/10555-0 - Stochastic modeling of interacting systems
• Grantee: Fabio Prates Machado
• Support Opportunities: Research Projects - Thematic Grants