树上有限字母模型的动态吉布斯-非吉布斯转变
Dynamical Gibbs-non-Gibbs transitions for finite-alphabet models on trees
浏览论文内容
中文总结 AI 辅助
该研究推广了树上有限字母模型的吉布斯性质转变结果，证明f-稳定吉布斯态下大时间可重返拟局域吉布斯区域，也给出f-鞍点时非拟局域性大时间持续存在的例证。
中文摘要 AI 辅助
我们研究树上的有限字母自旋模型，这类模型在独立对称自旋翻转动力学下演化。对于零外场下低温伊辛模型在格点上的演化加测度，已有研究表明，在所有足够大的时间下它都是非拟局域的（文献EnFeHoRe02）。然而在树上，伊辛模型的演化加相表现不同：拟局域吉布斯性质首先会丧失，但会在更晚的时间恢复（文献EnErIaKu12）。我们首先将该结果推广，证明对于所有有限字母模型，当动力学从f-稳定吉布斯态开始时，在树上的大时间下会更普遍地重新进入拟局域吉布斯区域。这类态通过描述模型的齐次递归f上的与时间无关的尖锐条件定义，可显式验证。作为第二个相反的结果，我们开发了一种方法，用于证明演化测度的非拟局域性在大时间下持续存在。我们表明，当初始吉布斯测度对应f-鞍点时，即使所有构型在大时间下都可能变为“坏”的，并对Potts模型给出了明确例证。在证明中，我们研究了空间非均匀扰动对时间依赖不动点问题解的影响。
英文摘要
We study finite-alphabet spin models on trees evolving under independent symmetric spin-flip dynamics. On the lattice the time-evolved plus measure of the low temperature Ising model in zero external field was shown to be non-quasilocal for all sufficiently large times in \cite{EnFeHoRe02}. On the tree however the time-evolved plus phase of the Ising model behaves differently, as the quasilocal Gibbs property is first lost, but then recovered at a later time \cite{EnErIaKu12}. We first extend this result by showing that large-time reentry into the quasilocal Gibbs regime on the tree happens more generally for all finite alphabet models, when the dynamics is started in $f$-stable Gibbs states. These are defined in terms of a time-independent sharp condition on the homogeneous recursion $f$ describing the model, which can be checked explicitly. As our second and opposite result, we develop a method for proving large-time persistence of non-quasilocality of time-evolved measures. We show that even all configurations can become bad at large times, when the initial Gibbs measure corresponds to an $f$-saddle and give an explicit illustration for the Potts model. In our proofs we study the influence of spatially inhomogeneous perturbations to the solutions of a time-dependent fixed point problem.
