arXiv 2107.045432021-07math.PR

具有耦合无序的完全图上Glauber动力学的亚稳性

Metastability for Glauber dynamics on the complete graph with coupling disorder

Anton Bovier, Frank den Hollander, Saeda Marello

备注：: 41 pages, 6 figures
发表信息：: Commun. Math. Phys. 392 (2022), 307-345
DOI：: 10.1007/s00220-022-04351-8

摘要：

考虑$n$顶点上的完全图。为每个顶点指定一个可以取值为$-1$或$+1$的Ising自旋。每个自旋$i \in [n]=\{1,2,\dots, n\}$与磁场$h \in [0,\infty)$相互作用，而每对自旋$i,j \in [n]$以耦合强度$n^{-1} J(i)J(j)$彼此相互作用，其中$J=(J(i))_{i \in [n]}$是i.i.d.从有限支持的概率分布中提取的非负随机变量。自旋翻转根据大都市动力学在相反的温度$β\in (0,\infty)$。我们表明，有关键的阈值$β_c$和$h_c(β)$，这样，在作为$n\to\infty$的限制，系统表现出亚稳态行为当且仅当$β\in (β_c, \infty)$和$h \in [0,h_c(β))$。我们的主要结果是一个尖锐的渐近性，乘性误差$1+o_n(1)$，从任何亚稳态的平均交叉时间的一组具有较低的自由能的状态。我们使用标准技术的潜在理论的方法来亚稳态。渐近中的首阶项不依赖于$J$的实现，而校正项则依赖于$J$的实现。校正项的主阶是$\sqrt{n}$乘以中心高斯随机变量，其复杂方差取决于$β,h$、$J$定律和亚稳态。临界阈值$β_c$和$h_c(β)$取决于$J$定律，亚稳态的数量也是如此。我们导出了$β_c$的一个显式公式，并确定了$β\mapsto h_c(β)$的一些性质。有趣的是，后者不一定是单调的，这意味着亚稳态交叉可能是可重入的。

英文摘要：

Consider the complete graph on $n$ vertices. To each vertex assign an Ising spin that can take the values $-1$ or $+1$. Each spin $i \in [n]=\{1,2,\dots, n\}$ interacts with a magnetic field $h \in [0,\infty)$, while each pair of spins $i,j \in [n]$ interact with each other at coupling strength $n^{-1} J(i)J(j)$, where $J=(J(i))_{i \in [n]}$ are i.i.d. non-negative random variables drawn from a probability distribution with finite support. Spins flip according to a Metropolis dynamics at inverse temperature $β\in (0,\infty)$. We show that there are critical thresholds $β_c$ and $h_c(β)$ such that, in the limit as $n\to\infty$, the system exhibits metastable behaviour if and only if $β\in (β_c, \infty)$ and $h \in [0,h_c(β))$. Our main result is a sharp asymptotics, up to a multiplicative error $1+o_n(1)$, of the average crossover time from any metastable state to the set of states with lower free energy. We use standard techniques of the potential-theoretic approach to metastability. The leading order term in the asymptotics does not depend on the realisation of $J$, while the correction terms do. The leading order of the correction term is $\sqrt{n}$ times a centred Gaussian random variable with a complicated variance depending on $β,h$, on the law of $J$ and on the metastable state. The critical thresholds $β_c$ and $h_c(β)$ depend on the law of $J$, and so does the number of metastable states. We derive an explicit formula for $β_c$ and identify some properties of $β\mapsto h_c(β)$. Interestingly, the latter is not necessarily monotone, meaning that the metastable crossover may be re-entrant.

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