Zero-Temperature Chaos in Bidimensional Models with Finite-Range Potentials

TL;DR

This work constructs a finite-range potential on the bidimensional full shift that exhibits zero-temperature chaos, extending the phenomenon known from higher dimensions to $d=2$ via an explicit Aubrun–Sablik simulation of a one-dimensional effectively closed subshift and a duplicating step. The authors define the potential as the indicator of a forbidden cylinder in a carefully engineered $2$D SFT that inherits two competing ground-state regimes, and they prove chaotic zero-temperature limits by establishing two a priori estimates: a bound on the growth of the reconstruction function and a bound on the relative complexity of the simulation extension. By orchestrating an alternating dominance between two subsystems with differing topological entropies, the sequence of equilibrium measures $\mu_{\beta}$ along even and odd temperatures concentrates on disjoint ground-state sets, preventing convergence as $\beta\to\infty$. The results connect to recursion-theoretic classifications of ground states and illustrate the rich low-temperature behavior possible even for finite-range potentials in two dimensions, with broader links to recent generalizations and alternative proofs in the literature.

Abstract

We construct a finite-range potential on a bidimensional full shift on a finite alphabet that exhibits a zero-temperature chaotic behavior as introduced by van Enter and Ruszel. This is the phenomenon where there exists a sequence of temperatures that converges to zero for which the whole set of equilibrium measures at these given temperatures oscillates between two sets of ground states. Brémont's work shows that the phenomenon of non-convergence does not exist for finite-range potentials in dimension one for finite alphabets; Leplaideur obtained a different proof for the same fact. Chazottes and Hochman provided the first example of non-convergence in higher dimensions $d\geq3$; we extend their result for $d=2$ and highlight the importance of two estimates of recursive nature that are crucial for this proof: the relative complexity and the reconstruction function of an extension. We note that a different proof of this result was found by Chazottes and Shinoda, at around the same time that this article was initially submitted and that a strong generalization has been found by Gayral, Sablik and Taati.

Figures (3)

• Figure 1: The biggest square is the square pattern $p\in \widetilde{\mathcal{A}}^{\llbracket 0,n\rrbracket^2}$. Let $u\in I(p,\ell'_k)$. The dashed square of size $2\ell'_k$ is a pattern in $\mathcal{L}(\widetilde{\widetilde{X}}, 2\ell'_k)$. The pattern located in the innermost square of size $\ell'_k$ belongs to $\widetilde{A}'_k \bigsqcup \widetilde{B}'_k$. The innermost dot represents $u+\tau'_k$.
• Figure 2: We represent the intermediate strip $J^*_{j,i}$ in $J_h$ that is obtained after discarding $J_g$ and $J_f$ constructed before.
• Figure 3: We represent a clustering of two squares of the kind $J_h$ that intersects on the right $J_g$ along their markers and on the left $J_f$ along their initial and terminal segments.

Theorems & Definitions (58)

• Theorem 1.1
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Definition 2.7
• Definition 2.8
• Theorem 2.9: Aubrun-Sablik AubrunSablik2013