Large deviations and concentration properties for ∇ϕ interface models
Large deviations and concentration properties for ∇ϕ interface models
复制标题
∇ψ 界面模型的大偏差和浓度特性
DOI：
10.1007/s004400050266
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发表时间：
2000
影响因子：
2
通讯作者：
D. Ioffe
中科院分区：
文献类型：
--
作者：
J. Deuschel;G. Giacomin;D. Ioffe
抽象的。我们考虑在dN≡D∩(ℤd/N(N∈ℤ+)外具有零边界条件的无质量场，D是ℝd的一个合适的子集，即ℙd/N上的具有哈密顿量的连续自旋吉布斯测度ℝℤN由H(ϕ)=∑x，y：|x−y|=1V(ϕ(X)−ϕ(Y))和ϕ(X)=0给出，对于x∈dnc。相互作用V被认为是严格凸且二阶导数有界的。这是(d+1)维界面的标准有效模型：ϕ表示界面在基本DN上的高度。由于基面尺度的选择，我们通过设置ξN=ϕ/N来以相同的因子缩放高度。研究了当离散化步长1/N趋于零(N→∞)时，随机曲面族{ξN}和诱导梯度场族∇NξN的各种浓缩和松弛性质。特别地，我们证明了{ξN}的一个大偏差原理，并证明了相应的速率函数由∫Dσ(∇u(X)dx给出，其中σ是模型的表面张力。这是一个多维版本的样本路径大偏差原理。我们利用这一结果研究了体积约束下ℙN的浓度性质，即(1/ND)∑x∈dNξN(X)保持在固定体积v>；0附近的约束和硬壁约束，即所有x的ξN(X)≥0。因此，这是一个体积为v的液滴位于硬壁上方的模型。我们证明，在这些约束下，重标度高度的场{ξN集中在涉及表面张力的变分问题的解上，正如相边界唯象理论所预测的那样。然而，我们的主要结果证明了梯度场{∇NξN(·)}到相应的极值吉布斯态的局部松弛性质。因此，我们的方法与传统的大偏差方法几乎没有共同点，在精神上更接近于水动力极限类型的论点。证明既有概率方面的，也有分析方面的。基本的分析工具是椭圆型方程的？P估计和杨氏测度理论。在概率工具方面，连续自旋系统的Helffer-Sjöstrand[31]PDE表示起了中心作用，我们根据随机环境中的随机游动以及T.Funaki和H.Spohn[25]最近关于梯度场结构的结果重写了它。
Abstract. We consider the massless field with zero boundary conditions outside DN≡D∩ (ℤd/N) (N∈ℤ+), D a suitable subset of ℝd, i.e. the continuous spin Gibbs measure ℙN on ℝℤd/N with Hamiltonian given by H(ϕ) = ∑x,y:|x−y|=1V(ϕ(x) −ϕ(y)) and ϕ(x) = 0 for x∈DNC. The interaction V is taken to be strictly convex and with bounded second derivative. This is a standard effective model for a (d + 1)-dimensional interface: ϕ represents the height of the interface over the base DN. Due to the choice of scaling of the base, we scale the height with the same factor by setting ξN = ϕ/N.We study various concentration and relaxation properties of the family of random surfaces {ξN} and of the induced family of gradient fields ∇NξN as the discretization step 1/N tends to zero (N→∞). In particular, we prove a large deviation principle for {ξN} and show that the corresponding rate function is given by ∫Dσ(∇u(x))dx, where σ is the surface tension of the model. This is a multidimensional version of the sample path large deviation principle. We use this result to study the concentration properties of ℙN under the volume constraint, i.e. the constraint that (1/Nd) ∑x∈DNξN (x) stays in a neighborhood of a fixed volume v > 0, and the hard–wall constraint, i.e. ξN (x) ≥ 0 for all x. This is therefore a model for a droplet of volume v lying above a hard wall. We prove that under these constraints the field {ξN of rescaled heights concentrates around the solution of a variational problem involving the surface tension, as it would be predicted by the phenomenological theory of phase boundaries. Our principal result, however, asserts local relaxation properties of the gradient field {∇NξN(·)} to the corresponding extremal Gibbs states. Thus, our approach has little in common with traditional large deviation techniques and is closer in spirit to hydrodynamic limit type of arguments. The proofs have both probabilistic and analytic aspects. Essential analytic tools are ?p estimates for elliptic equations and the theory of Young measures. On the side of probability tools, a central role is played by the Helffer–Sjöstrand [31] PDE representation for continuous spin systems which we rewrite in terms of random walk in random environment and by recent results of T. Funaki and H. Spohn [25] on the structure of gradient fields.