论文标题
对于晶格$ \ mathbb {z}^2 $的有限范围相互作用的平衡状态的零温度极限
On the absence of zero-temperature limit of equilibrium states for finite-range interactions on the lattice $\mathbb{Z}^2$
论文作者
论文摘要
我们在$ \ Mathcal {s}^{\ Mathbb {z}^2} $上构建有限范围的交互，其中$ \ m athcal {s} $是一个有限的集合，与之相关的平衡状态（即Shift-Invariant Gibbs States）无法汇聚为温度，因此无法汇聚。更确切地说，如果我们选择任何一个参数family $（μ_β）_ {β> 0} $，其中$μ_β$是这种交互作用的反向温度$β$的平衡状态，则不存在$ \ lim_ {β\ lim_ {β\ to \ infty}μ_β$。这解决了第一作者和Hochman提出的问题，他们在$ d \ geq 3 $，$ d $作为晶格的尺寸时获得了这种非对抗行为。
We construct finite-range interactions on $\mathcal{S}^{\mathbb{Z}^2}$, where $\mathcal{S}$ is a finite set, for which the associated equilibrium states (i.e., the shift-invariant Gibbs states) fail to converge as temperature goes to zero. More precisely, if we pick any one-parameter family $(μ_β)_{β>0}$ in which $μ_β$ is an equilibrium state at inverse temperature $ β$ for this interaction, then $\lim_{β\to\infty}μ_β$ does not exist. This settles a question posed by the first author and Hochman who obtained such a non-convergence behavior when $d\geq 3$, $d$ being the dimension of the lattice.
