论文标题
近似方法的亚稳定性：应用于不可逆的二维ISING和POTTS模型，没有外部字段
Approximation method to metastability: an application to non-reversible, two-dimensional Ising and Potts models without external fields
论文作者
论文摘要
当前研究的主要贡献是两倍。首先，我们研究了在低温方向上没有外部田间的有限二维晶格上的Ising和Potts模型的能量格局。这些模型的能量景观的完整分析是未知的，因为基态之间具有复杂的平稳鞍结构。我们完全根据梯形图的子树一组随机步行来完全表征这种结构。其次，我们提供了众所周知的潜在理论方法来能稳定性的简化。特别是，通过替换具有$ h^1 $ approximation的平衡潜力的变异原理的作用，例如Dirichlet和Thomson原理，我们开发了一种新方法，可以简单地应用于非可逆动力学。作为这种方法的应用，我们不仅分析了可逆的大都市束缚动力学的亚稳态行为，而且分析了与低温ising和Potts模型相关的几种有趣的非可逆动力学，并得出了Eyring-Kramers Law和Markov链的缩小这些模型的模型。
The main contribution of the current study is two-fold. First, we investigate the energy landscape of the Ising and Potts models on finite two-dimensional lattices without external fields in the low temperature regime. The complete analysis of the energy landscape of these models was unknown because of its complicated plateau saddle structure between the ground states. We characterize this structure completely in terms of a random walk on the set of sub-trees of a ladder graph. Second, we provide a considerable simplification of the well-known potential-theoretic approach to metastability. In particular, by replacing the role of variational principles such as the Dirichlet and Thomson principles with an $H^1$-approximation of the equilibrium potential, we develop a new method that can be applied to non-reversible dynamics as well in a simple manner. As an application of this method, we analyze metastable behavior of not only the reversible Metropolis-Hastings dynamics, but also of several interesting non-reversible dynamics associated with the low-temperature Ising and Potts models explained above, and derive the Eyring-Kramers law and the Markov chain model reduction of these models.
