Stochastic orders and the frog model

Abstract

The frog model starts with one active particle at the root of a graph and some number of dormant particles at all nonroot vertices. Active particles follow independent random paths, waking all inactive particles they encounter. We prove that certain frog model statistics are monotone in the initial configuration for two nonstandard stochastic dominance relations: the increasing concave and the probability generating function orders. This extends many canonical theorems. We connect recurrence for random initial configurations to recurrence for deterministic configurations. Also, the limiting shape of activated sites on the integer lattice respects both of these orders. Other implications include monotonicity results on transience of the frog model where the number of frogs per vertex decays away from the origin, on survival of the frog model with death, and on the time to visit a given vertex in any frog model.

20 pages; new French-language abstract; to appear in Annales de l'Institut Henri Poincaré

References in corpus (3)

• From transience to recurrence with Poisson tree frogs
• Recurrence for the frog model with drift on ${\mathbb{Z}}^d$
• The critical density for the frog model is the degree of the tree

Cited by in corpus (7)

• Infection spread for the frog model on trees
• Frogs on trees?
• Cover time for the frog model on trees
• Sensitivity of the frog model to initial conditions
• Diffusion-limited annihilating systems and the increasing convex order
• Deviation bounds for the first passage time in the frog model
• Active Phase for Activated Random Walks on the Lattice in all Dimensions