An upper bound for the critical probability of the frog model on homogeneous trees

Author(s)

Élcio Lebensztayn

• Document type: Doctoral Thesis
• Press: São Paulo.
• Institution: Universidade de São Paulo (USP). Instituto de Matemática e Estatística (IME/SBI)
• Defense date: 2005-08-18

Examining board members

Fabio Prates Machado; Oswaldo Scarpa Magalhães Alves; Luiz Renato Goncalves Fontes; Nancy Lopes Garcia; Marina Vachkovskaia

• Advisor: Serguei Popov; Fabio Prates Machado

Abstract

We study the frog model on the homogeneous tree, a discrete-time particle system whose dynamics is summarized next. Initially there is an independent and identically distributed random number of particles at each vertex of the tree; those placed at a fixed vertex are active, the others being inactive. Active particles perform independent discrete-time simple random walks, with probability of disappearance (1 - p) at each instant. An inactive particle becomes active once its vertex is hit by an active particle. We consider in this thesis the critical value p_c that separates the phase in which the process dies out almost surely from the phase in which there exist active particles at all times with positive probability. We prove an upper bound for the critical probability p_c, which improves the formerly known result for the case of one particle per vertex initial configuration. The employed argument builds on the description of the frog model as an oriented percolation model which dominates suitably defined branching processes. We also obtain the asymptotic value of the stated upper bound, showing that it equals the asymptotic value of the critical probability. (AU)