Pinning and wetting transition for (1+1)-dimensional fields with Laplacian interaction
Pinning and wetting transition for (1+1)-dimensional fields with Laplacian interaction
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具有拉普拉斯相互作用的 (1 1) 维场的钉扎和润湿转变
DOI：
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发表时间：
2007
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影响因子：
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通讯作者：
J. Deuschel
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作者：
F. Caravenna;J. Deuschel
我们考虑一个随机场$\varphi:\{1,...,N\}\to\mathbb{R}$作为吸引到缺陷线$\varphi=0$（即x轴）的线性链的模型。场的自由律是由密度$\exp(-\sum_iV(\Delta\varphi_i))$表示的关于$\mathbb{R}^N$上的勒贝格测度，其中$\Delta$是离散拉普拉斯函数我们允许很大的势类$V(\cdot)$。与缺陷线的交互作用是通过在每次接触x轴时给予领域一个奖励$\varepsilon\ge0$来引入的。我们称这个模型为钉住模型。我们考虑第二种模型，即润湿模型，在这种模型中，除了固定奖励外，场也被限制为非负。我们表明，这两个模型都经历了一个相变，因为固定奖励的强度$\varepsilon$不同：在固定（$a=\mathrm{p}$）和润湿（$a=\mathrm{w}$）的情况下，存在一个临界值$\varepsilon_c^a$，使得当$\varepsilon>\varepsilon_c^a$场接触缺陷线的正数次（局部化），而这不会发生在$\varepsilon<\varepsilon_c^a$（局部化）。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不可分析点。对于钉住模型，跃迁是二阶的，因此在$\varepsilon=\varepsilon_c^{\mathrm{p}}$处的场是离域的。另一方面，润湿模型中的过渡是一阶的，对于$\varepsilon=\varepsilon_c^{\mathrm{w}}$，场是局域的。我们方法的核心是对该领域的马尔可夫更新理论描述。
We consider a random field $\varphi:\{1,...,N\}\to\mathbb{R}$ as a model for a linear chain attracted to the defect line $\varphi=0$, that is, the x-axis. The free law of the field is specified by the density $\exp(-\sum_iV(\Delta\varphi_i))$ with respect to the Lebesgue measure on $\mathbb{R}^N$, where $\Delta$ 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.