Some Problems on Random Intervals and Annihilating Particles

Particles perform independent random walks on the integers, and are annihilated if they cross paths or land at the same point. The problem is to determine whether the origin is hit infinitely often. The answer is shown to depend on the initial distribution of particles in accordance with a "log log law ." Several equivalent models are mentioned. 1. An annihilating particle model. Let us start with a particle at each integer point on the line except 0. Let the particles perform independent simple random walks, moving a unit step to the right or left with probability 2 at each unit of time. If two particles cross each other's paths, or if they are about to land at a common point they "annihilate" each other (i .e ., are removed from the game before landing). QUESTION 1. IS P{the origin is ever hit} = 1 ? We conjecture that the answer is yes, but do not have a proof. One can consider a number of variants of the above model. The particles can move to the left or right with probabilities p # 1 and 1-p. The walk can be one sided, with particles taking a step to the left with probability p and remaining in place with probability 1-p. In this case annihilation occurs if a particle moves toward an occupied point, in which case both the occupying and moving particle are removed prior to the landing. The initial distribution of particles can also be varied. The question 1 is open in all these cases, but the model suggests another problem about which we can say something. 2. Two types of particles. Suppose we start with two types of particles, say black and white (b and w), arranged on some subset of the integers according to some initial distribution so that the colors alternate. Everything is as in the previous model, but now the probabilities of movement of the b and w particles are different. Thus, for example, in the one sided case white (resp. black) particles would move independently one step to the left with probability p (resp. q), and remain in place with probability 1-p (resp. 1-q). Annihilation of particles occurs exactly as before, but observe that it always takes place btween