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

TL;DR

This paper constructs a bidimensional finite-range potential demonstrating zero-temperature chaos, where equilibrium measures oscillate between ground states as temperature approaches zero, extending previous higher-dimensional results to two dimensions.

Contribution

It extends the phenomenon of zero-temperature chaos to two-dimensional finite-range potentials, highlighting key recursive estimates like relative complexity and reconstruction functions.

Findings

Constructed a 2D finite-range potential with zero-temperature chaos.

Extended non-convergence results from higher dimensions to 2D.

Identified crucial recursive estimates for the proof.

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.