论文标题
固体固体界面无序固定
Solid-On-Solid interfaces with disordered pinning
论文作者
论文摘要
我们研究了一个简单的界面模型的定位跃迁，该模型与非象内都是缺陷平面相互作用。该接口由函数$ ϕ的图建模：\ mathbb z^2 \ to \ mathbb z $，并且该疾病是通过固定的IID中心随机变量$（ω_x）_ {x \ in \ Mathbb z^2} $的固定实现的固定实现。系统的哈密致力于三个参数$α，β> 0 $和$ h \ in \ mathbb r $，它们分别确定最近邻居相互作用的强度为无序的振幅和与底物的相互作用的平均值，并且由表达式给出 $ \ MATHCAL H（ϕ）：=β\ sum_ {x \ sim y} | ϕ（x） - ϕ（y）| - \ sum_ {x}（αΩ_x+h）{\ bf 1} _ {\ bf 1} _ {\ {ϕ（x）= 0 \}} （SOS）模型。在该制度中，我们提供了从本地化阶段到离域的相应相关的一个相应的相应的相应和消失的点，以$ ϕ（x）= 0 $的积分分数提供了$ h $中的相转换的清晰描述。我们证明，$ h $的临界值与退火模型相对应，并由$ h_c（α）= - \ log \ mathbb e [e^{αΩ}] $给出，并且在临界点附近，自由能显示以下关键行为$$f_β（α，α，h_cc+u）\ sim} \ max_ {n \ ge 1} \ left \ {θ_1e^{ - 4βn} u- \ frac {1} {2}θ^θ^2_1 e^{ - 8βn} \ frac {\ frac {\ mathrm {var} e^{αΩ} \ right]^2} \ right \}。$$正常常数$θ_1（β）> 0 $由Infinite体积SOS的渐近概率定义为Infinite sos的渐近概率，$ 0 $ 0 $ bouncy bouncy条件$ bourdial条件$θ_1（β）：= \ lim_ = \ lim_ {n \ to \ for_ \ f to \ f tyb \ fty fty} e} （ϕ（{\ bf 0}）= n）$ ...
We investigate the localization transition for a simple model of interface which interacts with an inhomonegeous defect plane. The interface is modeled by the graph of a function $ϕ: \mathbb Z^2 \to \mathbb Z$,and the disorder is given by a fixed realization of a field of IID centered random variables$(ω_x)_{x\in \mathbb Z^2}$. The Hamiltonian of the system depends on three parameters $α,β>0$ and $h\in \mathbb R$ which determine respectively the intensity of nearest neighbor interaction the amplitude of disorder and the mean value of the interaction with the substrate, and is given by the expression $$\mathcal H(ϕ):= β\sum_{x\sim y} |ϕ(x)-ϕ(y)|- \sum_{x} (αω_x+h){\bf 1}_{\{ϕ(x)=0\}}.$$ We focus on the large-$β$/rigid phase phase of the Solid-On-Solid (SOS) model. In that regime, we provide a sharp description of the phase transition in $h$ from a localized phase to a delocalized one corresponding respectivelly to a positive and vanishing fraction of points with $ϕ(x)=0$. We prove that the critical value for $h$ corresponds to that of the annealed model and is given by $h_c(α)= -\log \mathbb E[e^{αω}]$, and that near the critical point, the free energy displays the following critical behavior $$F_β(α,h_c+u )\stackrel{u\to 0+}{\sim} \max_{n\ge 1} \left\{θ_1 e^{-4βn} u- \frac{1}{2}θ^2_1 e^{-8βn} \frac{\mathrm{Var}\left[e^{αω}\right]}{\mathbb E \left[ e^{αω} \right]^2}\right\}.$$ The positive constant $θ_1(β)>0$ is defined by the asymptotic probability of spikes for the infinite volume SOS with $0$ boundary condition $θ_1(β):=\lim_{n\to \infty} e^{4βn}\mathbf P_β (ϕ({\bf 0})=n)$ ...
