Metastability of the three-state Potts model with general interactions

IF 1.3 3区 数学 Q2 STATISTICS & PROBABILITY Electronic Journal of Probability Pub Date : 2023-01-01 DOI:10.1214/23-ejp1003
Gianmarco Bet, Anna Gallo, Seonwoo Kim
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引用次数: 5

Abstract

We consider the Potts model on a two-dimensional periodic rectangular lattice with general coupling constants Jij>0, where i,j∈{1,2,3} are the possible spin values (or colors). The resulting energy landscape is thus significantly more complex than in the original Ising or Potts models. The system evolves according to a Glauber-type spin-flipping dynamics. We focus on a region of the parameter space where there are two symmetric metastable states and a stable state, and the height of a direct path between the metastable states is equal to the height of a direct path between any metastable state and the stable state. We study the metastable transition time in probability and in expectation, the mixing time of the dynamics and the spectral gap of the system when the inverse temperature β tends to infinity. Then, we identify all the critical configurations that are visited with high probability during the metastable transition. Our main tool is the so-called pathwise approach to metastability, which requires a detailed analysis of the energy landscape.
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具有一般相互作用的三态波茨模型的亚稳态
考虑具有一般耦合常数Jij>0的二维周期矩形晶格上的Potts模型,其中i,j∈{1,2,3}是可能的自旋值(或颜色)。由此得出的能源格局比原来的伊辛或波茨模型要复杂得多。该系统根据格劳伯型自旋翻转动力学发展。我们关注参数空间中存在两个对称亚稳态和一个稳态的区域,并且亚稳态之间的直接路径的高度等于任意亚稳态与稳定状态之间的直接路径的高度。我们研究了在概率和期望上的亚稳态跃迁时间,动力学的混合时间和系统在温度逆β趋于无穷时的谱隙。然后,我们确定了在亚稳跃迁过程中高概率访问的所有关键配置。我们的主要工具是所谓的亚稳态路径方法,这需要对能量景观进行详细的分析。
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来源期刊
Electronic Journal of Probability
Electronic Journal of Probability 数学-统计学与概率论
CiteScore
1.80
自引率
7.10%
发文量
119
审稿时长
4-8 weeks
期刊介绍: The Electronic Journal of Probability publishes full-size research articles in probability theory. The Electronic Communications in Probability (ECP), a sister journal of EJP, publishes short notes and research announcements in probability theory. Both ECP and EJP are official journals of the Institute of Mathematical Statistics and the Bernoulli society.
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