Competing frogs on ${\mathbb Z}^{d}$

Abstract

A two-type version of the frog model on Z d is formulated, where active type i particles move according to lazy random walks with probability p i of jumping in each time step (i = 1, 2). Each site is independently assigned a random number of particles. At time 0, the particles at the origin are activated and assigned type 1 and the particles at one other site are activated and assigned type 2, while all other particles are sleeping. When an active type i particle moves to a new site, any sleeping particles there are activated and assigned type i, with an arbitrary tie-breaker deciding the type if the site is hit by particles of both types in the same time step. We show that the event G i that type i activates infinitely many particles has positive probability for all p 1 , p 2 ∈ (0, 1] (i = 1, 2). Furthermore, if p 1 = p 2 , then the types can coexist in the sense that P(G 1 ∩ G 2 ) > 0. We also formulate several open problems. For instance, we conjecture that, when the initial number of particles per site has a heavy tail, the types can coexist also when p 1 = p 2 .x ∈ Z d , j = 1, . . . , η(x)}. At time 0, the particles at the origin are activated, while all other particles are sleeping. When a particle is activated, it starts moving according to its associated random walk so that, in each time step, it moves to a uniformly chosen neighboring site. When a site is visited by an active particle, any sleeping particles at the site are activated and start moving. If the origin is non-empty, this means that the set of activated particles grows to infinity.The model has previously been studied e.g. with respect to transience/recurrence [25], the shape of the set of visited sites [2, 3] and extinction/survival for a version of the model including death of active particles [4]. Here we introduce a two-type version of the model, where an active particle can be of either of two types. We study the possibility for the types to activate infinitely many particles and investigate in particular the event of coexistence, which is said to occur if both types activate infinitely many particles. Similar questions have been studied for other competition models on Z d , for instance the so-called Richardson model where a site becomes type i infected (i = 1, 2) at a rate proportional to the number of nearest neighbor sites of type i. In our model however, the type is associated with the moving particles rather than the sites.

The initial set in the frog model is irrelevant

“…for large n. This was proved in [1] for a non-lazy process with η ≡ 1 and generalized to other initial distributions in [2]. The minor additions needed to get the result for a lazy process are described in [6]. The shape A inherits all symmetries of Z d and, since the growth occurs in discrete time, A cannot exceed the L 1 unit-ball.…”

“…Write G i for the event that type i activates infinitely many particles and C = G 1 ∩ G 2 for the event that the types coexist by doing so simultaneously. In [6] it is shown that, if either η(x) ≥ 1 almost surely or E[η(x)] < ∞ for any x ∈ Z d , then C has strictly positive probability when p 1 = p 2 ∈ (0, 1]. The condition on ν is of technical nature, and presumably not necessary for the conclusion.…”

“…Theorem 1.2 is restricted to bounded initial sets while, for unbounded initial sets, the choice of initial sets could potentially affect the possibility of coexistence; see [5] for related results for competing first passage percolation. For bounded initial sets, the possibility of coexistence is conjectured to be determined by the relation between the one-type shapes A(ν, p 1 ) and A(ν, p 2 ) of the two types; see [6]. This means that it would ECP 25 (2020), paper 50. be interesting to investigate how laziness affects the shape.…”

“…in [1,2,3]. The dynamics can also be based on lazy random walks, where a particle in a given time step independently performs a random walk jump with probability p or stays put with probability 1 − p; see [6].…”

“…Recently, a two-type version of the model was introduced in [6], where particles of type i move according to lazy random walks with jump probability p i . All particles in the initial i.i.d.…”