Planelike interfaces in long-range Ising models and connections with nonlocal minimal surfaces

TL;DR

This paper establishes the existence of planelike ground state solutions in long-range Ising models with nonlocal interactions and connects these solutions to nonlocal minimal surfaces through a rigorous limiting process.

Contribution

It constructs ground states with interfaces near hyperplanes for long-range Ising models and links these to nonlocal minimal surfaces, extending previous finite-range results.

Findings

Existence of planelike ground state solutions in long-range Ising models.

Construction of nonlocal minimal surfaces near hyperplanes.

A rigorous limit connecting long-range Ising models and nonlocal minimal surfaces.

Abstract

This paper contains three types of results: 1. the construction of ground state solutions for a long-range Ising model whose interfaces stay at a bounded distance from any given hyperplane, 2. the construction of nonlocal minimal surfaces which stay at a bounded distance from any given hyperplane, 3. the reciprocal approximation of ground states for long-range Ising models and nonlocal minimal surfaces. In particular, we establish the existence of ground state solutions for long-range Ising models with planelike interfaces, which possess scale invariant properties with respect to the periodicity size of the environment. The range of interaction of the Hamiltonian is not necessarily assumed to be finite and also polynomial tails are taken into account (i.e. particles can interact even if they are very far apart the one from the other). In addition, we provide a rigorous bridge between the theory of long-range Ising models and that of nonlocal minimal surfaces, via some precise limit result.