Local and Global Survival for Nonhomogeneous Random Walk Systems on Z

Abstract

Abstract. We study an interacting random walk system on Z where at time 0 there is an active particle at 0 and one inactive particle on each site n ≥ 1. Particles become active when hit by another active particle. Once activated, the particle starting at n performs an asymmetric, translation invariant, nearest neighbor random walk with left jump probability ln. We give conditions for global survival, local survival and infinite activation both in the case where all particles are immortal and in the case where particles have geometrically distributed lifespan (with parameter depending on the starting location of the particle). More precisely, once activated, the particle at n survives at each step with probability pn ∈ [0, 1]. In particular, in the immortal case, we prove a 0-1 law for the probability of local survival when all particles drift to the right. Besides that, we give sufficient conditions for local survival or local extinction when all particles drift to the left. In the mortal case, we provide sufficient conditions for global survival, local survival and local extinction (which apply to the immortal case with mixed drifts as well). Analysis of explicit examples is provided: we describe completely the phase diagram in the cases 1/2 − ln ∼ ±1/n α , pn = 1 and 1/2 − ln ∼ ±1/n α , 1 − pn ∼ 1/n β (where α, β > 0).

The nonhomogeneous frog model on ℤ

“…After this, 1.1 is used to obtain a formula (see Theorem 3.1) that provides a sharp condition distinguishing between transience and non-transience in the case where the X j 's are i.i.d. and which, for the particular case where X j = 1, builds on the result from [1] by giving a sharp result that supersedes the soft condition in (2) and, for the case where p j = 1 2 + C log j (for all but finitely many j), implies the existence of a phase transition at C = π 2 24 . Finally, 1.1 will also be employed to obtain a formula that builds on the result from [5] In order to move towards a proof of Theorem 1.1, we begin by defining the process {M j } where, for each j ≥ 1, M j represents the number of frogs originating in {0, 1, .…”

The frog model with drift on $\mathbb{R} $

“…From (1) it follows that f (t) = 2λlog(t + 1) represents a critical case with respect to transience vs. nontransience, in the sense that for f (t) = Clog(t+1) a value of C > 2λ implies non-transience and C < 2λ implies transience.…”

“…Another model which bears a (perhaps stronger) resemblance to the non-uniform model on Z was examined by Bertacchi, Machado, and Zucca in [1], where they looked at a frog model on Z that begins with an active frog at the origin and a single sleeping frog at each positive integer point, and where the frog originating at i, upon activation, performs an asymmetric random walk that goes left with probability p i and right with probability 1 − p i (i.e. the drift value can vary depending on the particular frog).…”

“…Due to this difference, activated frogs in the former case cannot be treated as interchangeable, as they are in the latter. Consequently, the methods used in [1]…”

“…For any t ≥ 0 let X t denote the number of frogs originating in [0, t] that ever pass the point x = t. Now note that lim t→∞ X t > 0 if and only if (i) the set of points (initially) containing sleeping frogs is unbounded (f being nonnegative and monotonically increasing implies it is bounded on compact sets, which guarantees that no bounded region will contain infinitely many sleeping frogs) and (ii) all of these frogs are eventually awakened. Since a Brownian motion with leftward drift 1 2 is continuous and goes to −∞ with probability 1, it follows that lim t→∞ X t > 0 ⇐⇒ {infinitely many frogs return to the origin}…”