论文标题
能量退化对一系列亚稳态状态的过渡时间的影响：应用于概率的细胞自动机
Effect of energy degeneracy on the transition time for a series of metastable states: application to Probabilistic Cellular Automata
论文作者
论文摘要
我们考虑具有指数较小的过渡概率的随机可逆动力学的亚竞服问题。我们将先前的结果推广到多个方向。我们对过渡矩阵的光谱间隙和相关动力学的混合时间进行了估计，以最大稳定性水平。这些与模型无关的结果尤其适用于大量概率的细胞自动机（PCA），然后我们将重点关注。我们在有限体积，小和固定的磁场以及消失的温度极限的情况下考虑PCA。该模型是特殊的，因为存在三个亚稳态，其中两个相对于它们的能量而退化。我们通过在其他所有配置的稳定性水平上给出明确的上限来严格识别亚稳态状态。我们依靠这些估计值证明了动力学的复发特性，这是途径稳定性方法的基石。此外，我们还根据潜在的亚稳定性理论方法来确定亚稳态状态，这使我们能够为从任何这样的亚稳态到稳定状态的预期过渡时间提供精确的渐近造型。
We consider the problem of metastability for stochastic reversible dynamics with exponentially small transition probabilities. We generalize previous results in several directions. We give an estimate of the spectral gap of the transition matrix and of the mixing time of the associated dynamics in terms of the maximal stability level. These model-independent results hold in particular for a large class of Probabilistic Cellular Automata (PCA), which we then focus on. We consider the PCA in a finite volume, at small and fixed magnetic field, and in the limit of vanishing temperature. This model is peculiar because of the presence of three metastable states, two of which are degenerate with respect to their energy. We identify rigorously the metastable states by giving explicit upper bounds on the stability level of every other configuration. We rely on these estimates to prove a recurrence property of the dynamics, which is a cornerstone of the pathwise approach to metastability. Further, we also identify the metastable states according to the potential-theoretic approach to metastability, and this allows us to give precise asymptotics for the expected transition time from any such metastable state to the stable state.
