论文标题
$ \ nablaϕ $接口模型和缩放限制的射线骑士定理
A Ray-Knight theorem for $\nablaϕ$ interface models and scaling limits
论文作者
论文摘要
我们引入了一种自然的措施，以在与梯度吉布斯量度相关的具有凸电势的梯度gibbs相关的langevin动力学驱动的时间依赖环境中演变而来的两种自然措施。我们得出了一个身份，该身份将该度量引起的泊松云的职业时间与相应的梯度场的平方相关，这通常不是高斯。在二次情况下，我们恢复了第二射线骑士定理的众所周知的概括。我们进一步确定了尺寸3中涉及的各种对象的缩放限制，这些对象被认为表现出均匀化。特别是，我们证明，梯度场的重新归一化的平方在适当的重新缩放下收敛到$ \ m \ m \ mathbb {r}^3 $上的高斯自由场的灯芯正方形，并具有合适的扩散矩阵，从而扩展了NADDAF和SPENCER在该领域本身的规模限制方面的著名结果。
We introduce a natural measure on bi-infinite random walk trajectories evolving in a time-dependent environment driven by the Langevin dynamics associated to a gradient Gibbs measure with convex potential. We derive an identity relating the occupation times of the Poissonian cloud induced by this measure to the square of the corresponding gradient field, which - generically - is not Gaussian. In the quadratic case, we recover a well-known generalization of the second Ray-Knight theorem. We further determine the scaling limits of the various objects involved in dimension 3, which are seen to exhibit homogenization. In particular, we prove that the renormalized square of the gradient field converges under appropriate rescaling to the Wick-ordered square of a Gaussian free field on $\mathbb{R}^3$ with suitable diffusion matrix, thus extending a celebrated result of Naddaf and Spencer regarding the scaling limit of the field itself.
