Compactness by coarse-graining in long-range lattice systems

TL;DR

This paper studies the limiting behavior of long-range lattice energies with coarse-graining, showing they converge to interfacial energies in a discrete-to-continuum limit, extending previous results for short-range interactions.

Contribution

It introduces a coarse-graining method for long-range lattice energies, demonstrating their convergence to surface energies, and extends coerciveness results to long-range interactions.

Findings

The energies converge to interfacial energies as the scale parameter tends to zero.

A coarse-graining procedure associates functions with sets of bounded perimeter.

The limit energy is explicitly computed for the integer lattice case.

Abstract

We consider energies on a periodic set ${\mathcal L}$ of ${\mathbb R}^d$ of the form $\sum_{i,j\in{\mathcal L}} a^\varepsilon_{ij}|u_i-u_j|$, defined on spin functions $u_i\in\{0,1\}$, and we suppose that the typical range of the interactions is $R_\varepsilon$ with $R_\varepsilon\to +\infty$, i.e., if $\|i-j\|\le R_\varepsilon$ then $a^\varepsilon_{ij}\ge c>0$. In a discrete-to-continuum analysis, we prove that the overall behaviour as $\varepsilon\to 0$ of such functionals is that of an interfacial energy. The proof is performed using a coarse-graining procedure which associates to scaled functions defined on $\varepsilon{\mathcal L}$ with equibounded energy a family of sets with equibounded perimeter. This agrees with the case of equibounded $R_\varepsilon$ and can be seen as an extension of coerciveness result for short-range interactions, but is different from that of other long-range interaction energies, whose limit exits the class of surface energies. A computation of the limit energy is performed in the case ${\mathcal L}={\mathbb Z}^d$.