Characterisation of the Set of Ground States of Uniformly Chaotic Finite-Range Lattice Models

Chaotic dependence on temperature refers to the phenomenon of divergence of Gibbs measures as the temperature approaches a certain value. Models with chaotic behaviour near zero temperature have multiple ground states, none of which are stable. We study the class of uniformly chaotic models, that is, those in which, as the temperature goes to zero, every choice of Gibbs measures accumulates on the entire set of ground states. We characterise the possible sets of ground states of uniformly chaotic finite-range models up to computable homeomorphisms. Namely, we show that the set of ground states of every model with finite-range and rational-valued interactions is topologically closed and connected, and belongs to the class $\Pi_2$ of the arithmetical hierarchy. Conversely, every $\Pi_2$-computable, topologically closed and connected set of probability measures can be encoded (via a computable homeomorphism) as the set of ground states of a uniformly chaotic two-dimensional model with finite-range rational-valued interactions.Comment: 46 pages, 12 figure

The Besicovitch-Stability of Noisy Tilings is Undecidable

International audienceIn this exploratory paper, I will first introduce a notion of stability, more lengthily described in a previous article. In this framework, I will then exhibit an unstable aperiodic tiling. Finally, by building upon the folkloric embedding of Turing machines into Robinson tilings, we will see that the question of stability is undecidable

Arithmetical Hierarchy of the Besicovitch-Stability of Noisy Tilings

37 pages, 8 figuresThe purpose of this article is to study the algorithmic complexity of the Besicovitch stability of noisy subshifts of finite type, a notion studied in a previous article. First, we exhibit an unstable aperiodic tiling, and then see how it can serve as a building block to implement several reductions from classical undecidable problems on Turing machines. It will follow that the question of stability of subshifts of finite type is undecidable, and the strongest lower bound we obtain in the arithmetical hierarchy is $\Pi_2$-hardness. Lastly, we prove that this decision problem, which requires to quantify over an uncountable set of probability measures, has a $\Pi_4$ upper bound

Arithmetical Hierarchy of the Besicovitch-Stability of Noisy Tilings

37 pages, 8 figuresThe purpose of this article is to study the algorithmic complexity of the Besicovitch stability of noisy subshifts of finite type, a notion studied in a previous article. First, we exhibit an unstable aperiodic tiling, and then see how it can serve as a building block to implement several reductions from classical undecidable problems on Turing machines. It will follow that the question of stability of subshifts of finite type is undecidable, and the strongest lower bound we obtain in the arithmetical hierarchy is $\Pi_2$-hardness. Lastly, we prove that this decision problem, which requires to quantify over an uncountable set of probability measures, has a $\Pi_4$ upper bound

Arithmetical Hierarchy of the Besicovitch-Stability of Noisy Tilings

37 pages, 8 figuresThe purpose of this article is to study the algorithmic complexity of the Besicovitch stability of noisy subshifts of finite type, a notion studied in a previous article. First, we exhibit an unstable aperiodic tiling, and then see how it can serve as a building block to implement several reductions from classical undecidable problems on Turing machines. It will follow that the question of stability of subshifts of finite type is undecidable, and the strongest lower bound we obtain in the arithmetical hierarchy is $\Pi_2$-hardness. Lastly, we prove that this decision problem, which requires to quantify over an uncountable set of probability measures, has a $\Pi_4$ upper bound

On the Besicovitch-Stability of Noisy Random Tilings

The v2 fixes a few typos, as well as a small numerical mistake in the last theorem. 35 pages, 7 figuresIn this paper, we introduce a noisy framework for SFTs, allowing some amount of forbidden patterns to appear. Using the Besicovitch distance, which permits a global comparison of configurations, we then study the closeness of noisy measures to non-noisy ones as the amount of noise goes to 0. Our first main result is the full classification of the (in)stability in the one-dimensional case. Our second main result is a stability property under Bernoulli noise for higher-dimensional periodic SFTs, which we finally extend to an aperiodic example through a variant of the Robinson tiling