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Zero-Temperature Chaos in Bidimensional Models with Finite-Range Potentials
Dynamical Systems · Submitted 22 Aug 2022
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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The abstract, sentence by sentence
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. Br\'emont'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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | contribution 0.95 | 0.95 | 0.80 | asserted 1.00 |
| s1 | 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. | background 0.58 | 0.09 | 0.11 | asserted 1.00 |
| s2 | Br\'emont'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. | background 0.98 | 0.03 | 0.04 | asserted 1.00 |
| s3 | 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. | contribution 0.97 | 0.90 | 0.70 | asserted 1.00 |
| s4 | 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. | background 0.95 | 0.10 | 0.05 | asserted 1.00 |
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The paper
Original 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. Br\'emont'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.
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1 arXiv paper cite this in 1 sentence
Papers whose main result relates to this one's
States the same result 1 paper
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In this work we present a construction of a bidimensional example described by a finite alphabet and a locally constant potential in which there exists a subsequence $(\beta_k)_{k\geq 0}$ where the non-convergence occurs for any sequence of equilibrium measures at inverse of temperature $\beta_k$ when $\beta_k\rightarrow+\infty$.
Chaos and Turing Machines on Bidimensional Models at Zero Temperature · 0.89
What it cites · 2 sentences without a confident label
Compares with 2 papers
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A preliminary version was submitted to arxiv [ref] at about the same time when a paper of Chazottes and Shinoda [ref] was submitted proving the same result but with a different proof.
On the absence of zero-temperature limit of equilibrium states for finite-range interactions on the lattice $\mathbb{Z}^2$ · Introduction · 0.96
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A preliminary version was submitted to arxiv [ref] at about the same time when a paper of Chazottes and Shinoda [ref] was submitted proving the same result but with a different proof.
Chaos and Turing Machines on Bidimensional Models at Zero Temperature · Introduction · 0.96
Cites as background 1 paper
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A discussion about when the several notions of Gibbsianess are equivalent (or not) can be found in [ref] .
Gibbs measures on Subshifts · Introduction · 0.99
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