Non-Gibbsianness of the invariant measures of non-reversible cellular automata with totally asymmetric noise

Abstract

We present a class of random cellular automata with multiple invariant measures which are all non-Gibbsian. The automata have configuration space {0,1}^{Z^d}, with d > 1, and they are noisy versions of automata with the "eroder property". The noise is totally asymmetric in the sense that it allows random flippings of "0" into "1" but not the converse. We prove that all invariant measures assign to the event "a sphere with a large radius L is filled with ones" a probability \mu_L that is too large for the measure to be Gibbsian. For example, for the NEC automaton -ln(\mu_L) ~ L while for any Gibbs measure the corresponding value is ~ L^2.