[Paper Review] Strict Convexity of the Surface Tension for Non-convex Potentials

Stefan Adams, Roman Kotecký+1 more|arXiv (Cornell University)|Jun 30, 2016

Stochastic processes and statistical mechanics12 citations

TL;DR

This paper establishes the strict convexity of the surface tension in lattice gradient models with non-convex interactions at low temperatures and small interface tilts, using multi-scale renormalization group techniques. It extends prior results by Funaki and Spohn—which required strictly convex potentials—to non-convex microscopic interactions, proving that surface tension remains strictly convex despite loss of convexity in the interaction potential.

ABSTRACT

We study gradient models on the lattice $\mathbb{Z}^d$ with non-convex interactions. These Gibbs fields (lattice models with continuous spin) emerge in various branches of physics and mathematics. In quantum field theory they appear as massless field theories. Even though our motivation stems from considering vector valued fields as displacements for atoms of crystal structures and the study of the Cauchy-Born rule for these models, our attention here is mostly devoted to interfaces, with the gradient field as an \emph{effective} interface interaction. In this case we prove the strict convexity of the surface tension (interface free energy) for low temperatures and sufficiently small interface tilts using muli-scale (renormalisation group analysis) techniques following the approach of Brydges and coworkers \cite{B07}. This is a complement to the study of the high temperature regime in \cite{CDM09} and it is an extension of Funaki and Spohn's result \cite{FS97} valid for strictly convex interactions.

Motivation & Objective

• To establish uniform convexity properties of surface tension in lattice gradient models with non-convex microscopic interactions.
• To extend rigorous renormalization group techniques beyond models with discrete rotational symmetry.
• To prove strict convexity of the surface tension for low-temperature, small-tilt interfaces in non-convex gradient models.
• To complement high-temperature results from [CDM09] by analyzing the low-temperature regime.
• To generalize Funaki and Spohn’s result on strictly convex interactions to non-convex potentials using multi-scale analysis.

Proposed method

• Employs multi-scale (renormalization group) analysis following Brydges’ framework [Bry09] to handle non-convex interactions.
• Uses finite-range decomposition of the Gibbs measure to iteratively coarse-grain the system at different scales.
• Defines polymer functionals and ideal Hamiltonians to represent effective interactions at each scale.
• Applies a renormalization transformation $ T_k: (H_k, K_k) o (H_{k+1}, K_{k+1}) $ to track changes in interaction terms across scales.
• Implements fine-tuning of initial conditions to ensure convergence and strict convexity in the thermodynamic limit.
• Applies implicit function theorems with loss of regularity to control differentiability and analyticity of the surface tension.

Experimental results

Research questions

• RQ1 Does the surface tension remain strictly convex in gradient models with non-convex interactions at low temperatures?
• RQ2 Can multi-scale renormalization group techniques be extended to models lacking discrete rotational symmetry?
• RQ3 How does the loss of convexity in the interaction potential affect the differentiability and convexity of the surface tension?
• RQ4 What conditions on the interaction potential and temperature ensure strict convexity of the surface tension?
• RQ5 Can the implicit function theorem with loss of regularity be used to prove smoothness and strict convexity of the surface tension in non-convex models?

Key findings

• The surface tension is strictly convex for low temperatures and sufficiently small interface tilts, even when the interaction potential $ W $ is non-convex.
• The strict convexity is established via a multi-scale renormalization group approach that tracks the evolution of interaction terms across scales.
• The method overcomes the lack of symmetry and non-convexity by using fine-tuning of initial conditions and loss-of-regularity implicit function theorems.
• The result generalizes Funaki and Spohn’s theorem from strictly convex to non-convex potentials, extending its applicability to more realistic models.
• The surface tension is shown to be $ C^m $-smooth for all $ m $, with continuous Peano derivatives up to order $ m $, ensuring strict convexity.
• The analysis confirms that the surface tension remains strictly convex even when the microscopic potential $ V $ is non-convex, provided $ \beta $ is large and tilt is small.