Stochastic orders and the frog model

TL;DR

This paper investigates how certain stochastic orders influence the behavior of the frog model, revealing monotonicity properties and extending classical theorems in the context of activated sites and recurrence.

Contribution

It establishes monotonicity of frog model statistics under nonstandard stochastic dominance relations, connecting initial configurations to model behavior and extending existing theorems.

Findings

Monotonicity of frog model statistics under concave and probability generating function orders.

Connection between recurrence in random and deterministic initial configurations.

Results on the shape and transience of the frog model under various initial conditions.

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.