Cover time for the frog model on trees

Abstract

The frog model is a branching random walk on a graph in which particles branch only at unvisited sites. Consider an initial particle density of $\mu$ on the full $d$-ary tree of height $n$. If $\mu =\Omega (d^2)$, all of the vertices are visited in time $\Theta (n\log n)$ with high probability. Conversely, if $\mu =O(d)$ the cover time is $\exp (\Theta (\sqrt{n}))$ with high probability.

arXiv comment36 pages; revisions in response to referees' comments; accepted in Forum of Math Sigma, Probability

References in corpus (10)

• Paths to Self-Organized Criticality
• From transience to recurrence with Poisson tree frogs
• Recurrence and transience for the frog model on trees
• The shape theorem for the frog model with random initial configuration
• Stochastic orders and the frog model
• The critical density for the frog model is the degree of the tree
• Infection spread for the frog model on trees
• Frogs on trees?
• Rapid social connectivity
• Universality and Sharpness in Absorbing-State Phase Transitions

Cited by in corpus (3)

• A new upper bound for the critical probability of the frog model on homogeneous trees
• Infection spread for the frog model on trees
• Laws of large numbers for the frog model on the complete graph