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**1 - 3**of**3**### [1] Non-Gibbsianness of the invariant measures of non-reversible cellular automata with totally asymmetric noise

, 2001

"... We present a class of random cellular automata with multiple invariant measures which are all d non-Gibbsian. The automata have configuration space {0,1} Z, 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 r ..."

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We present a class of random cellular automata with multiple invariant measures which are all d non-Gibbsian. The automata have configuration space {0,1} Z, 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 µL that is too large for the measure to be Gibbsian. For example, for the NEC automaton ( − ln µL) ≍ L while for any Gibbs measure the corresponding value is ≍ L2. Key words: Gibbs vs. non-Gibbs measures, cellular automata, invariant measures, non-ergodicity, eroders, convex sets.

### Toward a mathematical theory of renormalization

"... Renormalization transformations were developed by theoretical physicists in order to investigate first problems arising in quantum field theory and later in statistical mechanics, specifically phase transitions and critical phenomena appearing in systems of a large number of interacting components. ..."

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Renormalization transformations were developed by theoretical physicists in order to investigate first problems arising in quantum field theory and later in statistical mechanics, specifically phase transitions and critical phenomena appearing in systems of a large number of interacting components. In their latter version they provide a scheme of systematic