Non-Ergodicity in a 1-D Particle Process with Variable Length

Abstract

We present a 1-D random particle process with uniform local interaction, which displays some form of non-ergodicity, similar to contact processes, but more unexpected. Particles, enumerated by integer numbers, interact at every step of the discrete time only with their nearest neighbors. Every particle has two possible states, called minus and plus. At every time step two transformations occur. The first one turns every minus into plus with probability β independently from what happens at other places and thereby favors pluses against minuses. The second one is “impartial.” Under its action, whenever a plus is a left neighbor of a minus, both disappear with probability α independently from presence and fate of other pairs of this sort. If β is small enough by comparison with α ² and we start with “all minuses,” the minuses never die out.