具有Laplacian相互作用的（1+1）维场的钉扎和润湿转变
Pinning and wetting transition for (1+1)-dimensional fields with Laplacian interaction
摘要：
我们考虑一个随机场$φ:\{1,...,N\}\to\mathbb{R}$作为一个模型的线性链吸引到缺陷线$φ=0$，即，x轴。场的自由律由相对于$\mathbb{R}^N$上的勒贝格测度的密度$\exp(-\sum_iV(Δφ_i))$指定，其中$Δ$是离散拉普拉斯算子，并且我们允许非常大的一类势$V(\cdot)$。与缺陷线的相互作用是通过在场每次接触x轴时给予场奖励$\varepsilon\ge0$来引入的。我们称这个模型为钉扎模型。我们考虑第二个模型，润湿模型，其中，除了钉扎奖励，该字段也被约束为保持非负。我们发现，这两种模式进行相变的强度$\varepsilon$的钉扎奖励的变化：无论是在钉扎（$a=\mathrm{p}$）和润湿（$a=\mathrm{w}$）的情况下，存在一个临界值$\varepsilon_c^a$，使当$\varepsilon>\varepsilon_c^a$的字段接触缺陷线的正分数的时间（本地化），而这不会发生$\varepsilon<\varepsilon_c^a$（离域）。这两个临界值是非平凡的和不同的：$0<\varepsilon_c^{\mat hrm{p}}<\varepsilon_c^{\mathrm{w}}<\infty$，它们是各自自由能的唯一非解析点。对于钉扎模型，跃迁是二阶的，因此在$\varepsilon=\varepsilon_c^{\mathrm{p}}$的场是离域的。另一方面，在润湿模型的过渡是一阶和$\varepsilon=\varepsilon_c^{\mathrm{w}}$的字段是本地化的。我们的方法的核心是一个马尔可夫更新理论描述的领域。
英文摘要：
We consider a random field $φ:\{1,...,N\}\to\mathbb{R}$ as a model for a linear chain attracted to the defect line $φ=0$, that is, the x-axis. The free law of the field is specified by the density $\exp(-\sum_iV(Δφ_i))$ with respect to the Lebesgue measure on $\mathbb{R}^N$, where $Δ$ is the discrete Laplacian and we allow for a very large class of potentials $V(\cdot)$. The interaction with the defect line is introduced by giving the field a reward $\varepsilon\ge0$ each time it touches the x-axis. We call this model the pinning model. We consider a second model, the wetting model, in which, in addition to the pinning reward, the field is also constrained to stay nonnegative. We show that both models undergo a phase transition as the intensity $\varepsilon$ of the pinning reward varies: both in the pinning ($a=\mathrm{p}$) and in the wetting ($a=\mathrm{w}$) case, there exists a critical value $\varepsilon_c^a$ such that when $\varepsilon>\varepsilon_c^a$ the field touches the defect line a positive fraction of times (localization), while this does not happen for $\varepsilon<\varepsilon_c^a$ (delocalization). The two critical values are nontrivial and distinct: $0<\varepsilon_c^{\mat hrm{p}}<\varepsilon_c^{\mathrm{w}}<\infty$, and they are the only nonanalyticity points of the respective free energies. For the pinning model the transition is of second order, hence the field at $\varepsilon=\varepsilon_c^{\mathrm{p}}$ is delocalized. On the other hand, the transition in the wetting model is of first order and for $\varepsilon=\varepsilon_c^{\mathrm{w}}$ the field is localized. The core of our approach is a Markov renewal theory description of the field.
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