[Paper Review] Non-Gibbsianness of the invariant measures of non-reversible cellular automata with totally asymmetric noise

Roberto Fernández, André Toom|ArXiv.org|Jan 15, 2001

Cellular Automata and Applications22 references10 citations

TL;DR

This paper demonstrates that invariant measures of non-reversible cellular automata with totally asymmetric noise—where 0s can stochastically flip to 1s but not vice versa—are non-Gibbsian. Using a damage-spreading mechanism in eroder-type automata on d≥2 lattices, it shows that large uniform regions of 1s have exponentially too high probability, violating Gibbsianness as evidenced by sub-volume exponential decay of the log-probability of spherical regions of size L.

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 μ_L that is too large for the measure to be Gibbsian. For example, for the NEC automaton -ln(μ_L) ~ L while for any Gibbs measure the corresponding value is ~ L^2.

Motivation & Objective

• To investigate the Gibbsianness of invariant measures in non-reversible stochastic cellular automata with asymmetric noise.
• To determine whether such measures can correspond to equilibrium states of statistical mechanical systems.
• To establish that multiple invariant measures in this class are non-Gibbsian despite being stationary under the dynamics.
• To extend understanding of non-Gibbsianness beyond cases with symmetric noise or detailed balance.

Proposed method

• Constructs a class of stochastic cellular automata on {0,1}^Z^d with d>1, derived from deterministic eroder automata with one-sided noise (0→1 only).
• Analyzes the dynamics via a 'spider' or initial seed configuration that propagates to fill large spheres of 1s over time.
• Uses a lower bound on the probability of observing a sphere of 1s of radius L, showing it decays as exp(−cL) rather than the Gibbsian exp(−cL²).
• Applies renormalization and block transformation arguments to show non-Gibbsianness is preserved under single-site transformations.
• Employs minoration techniques to extend results to broader classes of random cellular automata satisfying certain transition rate conditions.
• Considers specific examples such as the NEC model and generalizes to functions f with varying sensitivity to coordinates, deriving stronger bounds for certain f.

Experimental results

Research questions

• RQ1 Do invariant measures of non-reversible cellular automata with totally asymmetric noise exhibit Gibbsianness?
• RQ2 Can large uniform clusters of 1s in such systems have probabilities inconsistent with Gibbs measures?
• RQ3 Is the non-Gibbsianness of these measures robust under renormalization transformations?
• RQ4 Does the absence of detailed balance and the presence of one-way noise lead to non-Gibbsianness even when multiple invariant measures exist?
• RQ5 Can the decay rate of the probability of large uniform regions be used as a diagnostic for non-Gibbsianness?

Key findings

• All invariant measures of the studied class of non-reversible cellular automata with totally asymmetric noise are non-Gibbsian.
• The probability μ_L of a sphere of radius L being filled with 1s decays as exp(−cL), which is too fast for Gibbsianness, since Gibbs measures require exp(−cL²) decay.
• For the NEC automaton, −ln μ_L ≍ L, while any Gibbs measure would require −ln μ_L ≍ L², demonstrating a clear violation of Gibbsian behavior.
• The non-Gibbsianness arises from a damage-spreading mechanism: a single 'spider' of 1s can grow to fill a large sphere, making such configurations overly probable.
• The result holds even when the dynamics are not reversible and do not satisfy detailed balance, and extends to broader classes of random cellular automata satisfying minimal transition rate conditions.
• The mechanism of non-Gibbsianness is robust under single-site renormalization, implying that the measures cannot arise from block averaging of Gibbsian measures.