We consider the problem of metastability for stochastic dynamics with exponentially small transition probabilities in the low temperature limit. We generalize previous model-independent results in several directions. First, we give an estimate of the mixing time of the dynamics in terms of the maximal stability level. Second, assuming the dynamics is reversible, we give an estimate of the associated spectral gap. Third, we give precise asymptotics for the expected transition time from any metastable state to the stable state using potential-theoretic techniques. We do this in a general reversible setting where two or more metastable states are allowed and some of them may even be degenerate. This generalizes previous results that hold for a series of only two metastable states. We then focus on a specific Probabilistic Cellular Automata (PCA) with configuration space \({\mathcal {X}}=\{-1,+1\}^\varLambda \) where \(\varLambda \subset {\mathbb {Z}}^2\) is a finite box with periodic boundary conditions. We apply our model-independent results to find sharp estimates for the expected transition time from any metastable state in \(\{\underline{-1}, {\underline{c}}^o,{\underline{c}}^e\}\) to the stable state \(\underline{+1}\). Here \({\underline{c}}^o,{\underline{c}}^e\) denote the odd and the even chessboard respectively. To do this, 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.

我们考虑了在低温极限下具有指数级小跃迁概率的随机动力学的亚稳态问题。我们在几个方向上概括了以前与模型无关的结果。首先，我们根据最大稳定水平给出动力学混合时间的估计。其次，假设动力学是可逆的，我们给出相关光谱间隙的估计。第三，我们使用势论技术给出了从任何亚稳态到稳定状态的预期过渡时间的精确渐近线。我们在一般可逆环境中这样做，其中允许两个或多个亚稳态，其中一些甚至可能退化。这概括了先前的结果，这些结果仅适用于一系列只有两个亚稳态。\({\mathcal {X}}=\{-1,+1\}^\varLambda \)其中\(\varLambda \subset {\mathbb {Z}}^2\)是具有周期性边界条件的有限框. 我们应用我们的模型独立结果来找到对\(\{\underline{-1}, {\underline{c}}^o,{\underline{c}}^ e\}\)到稳定状态\(\underline{+1}\)。这里\({\underline{c}}^o,{\underline{c}}^e\)分别表示奇数和偶数棋盘。为此，我们通过给出每个其他配置的稳定性水平的明确上限来严格识别亚稳态。我们依靠这些估计来证明动力学的重复特性，这是亚稳态路径方法的基石。