Quantitative homogenization of the disordered ∇ ϕ model

2018/10/15 by Paul Dario, Dario, Paul
Computer Science · Physics and Astronomy · Mathematics · #Advanced Mathematical Modeling in Engineering #Theoretical and Computational Physics #Stochastic processes and statistical mechanics

paper · doi:10.48550/arxiv.1810.06428

Abstract

We study the ∇ ϕ model with uniformly convex Hamiltonian H (ϕ) := ∑ V(∇ ϕ) and prove a quantitative rate of convergence for the finite-volume surface tension as well as a quantitative rate estimate for the L²-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-Sjöstrand 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.

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