Abstract
We study the $\nabla \phi$ model with uniformly convex Hamiltonian $\mathcal{H} (\phi) := \sum V(\nabla \phi)$ and prove a quantitative rate of convergence for the finite-volume surface tension as well as a quantitative rate estimate for the $L^2$-norm for the field subject to affine boundary condition. One of our motivations is to develop a new toolbox for studying this problem that does not rely on the Helffer-Sj\ostrand representation. Instead, we make use of the variational formulation of the partition function, the notion of displacement convexity from the theory of optimal transport, and the recently developed theory of quantitative stochastic homogenization.
Highlights
Many physical phenomena exhibit a transition between two pure phases, especially at low temperature
The first mathematical model to understand the macroscopic shape of the interface separating the two phases was introduced by Wulff in 1901 in [45] to describe the shape of a crystal at equilibrium: it characterizes the interfaces as minimizers of the Wulff functional, defined by, for a subset E ⊆ Rd, W (E) := σ (n(x)) dx
Many important results in this direction were obtained on various models in the 90s; in [1], Alexander, Chayes and Chayes derived a Wulff construction for the two dimensional supercritical Bernoulli bond percolation
Summary
Many physical phenomena exhibit a transition between two pure phases, especially at low temperature. A first reasonable objective would be to extend these results to more general conditions, instead of the affine boundary condition presented in Theorem 1.2 Such a result would prove that the properly rescaled version of the interface φ converges quantitatively to a deterministic interface, which is a critical point of the Wulff functional involving the surface tension ν. While most of the theory developed to understand stochastic homogenization focuses on linear elliptic equations, the closest analogy with the ∇φ interface model is the stochastic homogenization of nonlinear equations In this setting, the results are more sparse: one can mention the work of Armstrong, Mourrat and Smart [7, 8] who quantified the works of Dal Maso and Modica [18, 19]. In particular (z + n)z∈Zn is a partition of n+1
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