[Paper Review] Scaling limit and convergence of smoothed covariance for gradient models with non-convex potential

Susanne Hilger|arXiv (Cornell University)|Mar 15, 2016

Theoretical and Computational Physics9 references3 citations

TL;DR

This paper establishes the scaling limit of discrete gradient models with non-convex potentials as a continuum massless Gaussian free field, using a renormalization group representation from [AKM]. It proves a Central Limit Theorem for strongly dependent fields and demonstrates convergence of smoothed covariances, extending Gaussian behavior to non-convex interactions under small perturbations of the quadratic potential.

ABSTRACT

A discrete gradient model for interfaces is studied. The interaction potential is a non-convex perturbation of the quadratic gradient potential. Based on a representation for the finite volume Gibbs measure obtained via a renormalization group analysis by Adams, Kotecký and Müller in [AKM] it is proven that the scaling limit is a continuum massless Gaussian free field. From probabilistic point of view, this is a Central Limit Theorem for strongly dependent random fields. Additionally, the convergence of covariances, smoothed on a scale smaller than the system size, is proven.

Motivation & Objective

• To establish the scaling limit of discrete gradient models with non-convex potentials as a continuum massless Gaussian free field.
• To prove a Central Limit Theorem for strongly dependent random fields arising in such models.
• To demonstrate convergence of covariances smoothed on a scale smaller than the system size.
• To extend known Gaussian behavior in convex models to non-convex perturbations using the renormalization group representation from [AKM].
• To analyze the long-distance behavior of gradient Gibbs measures under small non-convex perturbations of the quadratic interaction.

Proposed method

• Uses a finite-volume Gibbs measure representation derived from renormalization group analysis in [AKM] for non-convex gradient models.
• Implements periodic boundary conditions and shifted potentials to maintain translation invariance, replacing Dirichlet conditions.
• Applies a torus-based lattice structure with side length $ L^N $, ensuring periodicity and enabling scaling limits.
• Employs weak convergence techniques in $ L^2 $ and $ L^2_{ ext{loc}} $ to extract limit fields from sequences of discrete gradients.
• Uses discrete Poincaré inequalities and diagonal sequence arguments to establish convergence of fields and their gradients.
• Applies estimates on large field regulators and norm bounds on $ abla^s ilde{ abla}^* ilde{C}^q $ to control error terms in the scaling limit.

Experimental results

Research questions

• RQ1 Does the scaling limit of a gradient model with a non-convex potential still converge to a massless Gaussian free field?
• RQ2 How do covariances behave under smoothing at scales smaller than the system size in such models?
• RQ3 Can a Central Limit Theorem be established for strongly dependent random fields arising from non-convex gradient interactions?
• RQ4 What is the long-distance behavior of the Gibbs measure when the interaction potential is a small non-convex perturbation of the quadratic potential?
• RQ5 Does the renormalization group representation from [AKM] allow for convergence results beyond the surface tension to the Gaussian free field?

Key findings

• The scaling limit of the discrete gradient model with non-convex potential converges weakly to a continuum massless Gaussian free field.
• The model satisfies a Central Limit Theorem for strongly dependent random fields, extending classical CLT results to non-Gaussian, non-convex interactions.
• Smoothed covariances converge as the system size increases, with convergence rates controlled by $ au( heta) = L^{(1- heta)(d/2 + 4)} $.
• The limit field arises from weak convergence of discrete gradients $ D_N u_N $ to a field $ v = D u $ in $ L^2_{ ext{loc}}(\mathbb{R}^d) $, with $ u $ solving a limiting PDE.
• Error terms in the renormalization group expansion are uniformly bounded via large field regulator estimates, ensuring convergence under small non-convex perturbations.
• The convergence of the Gibbs measure to the Gaussian free field holds under smallness conditions on $ \sqrt{\beta}\|g\|_{L^1} $, even when the full potential $ W $ is non-convex.