Quasicrystal problem -- on rigidity of non-periodic structures from statistical mechanics point of view

TL;DR

This paper explores the stability of non-periodic quasicrystal structures using statistical mechanics, emphasizing the importance of homogeneity conditions for their robustness against perturbations.

Contribution

It introduces the concept of the Strict Boundary Condition as essential for the stability of non-periodic ground states in quasicrystals.

Findings

Homogeneity is crucial for stability of non-periodic structures.

Strict Boundary Condition ensures robustness against fluctuations.

Ground states of certain Hamiltonians correspond to quasicrystal tilings.

Abstract

We present a brief history of quasicrystals and a short introduction to classical lattice-gas models of interacting particles. We discuss stability of non-periodic tilings and one-dimensional sequences of symbols seen as ground states of some hamiltonians. We argue that some sort of homogeneity, the so-called Strict Boundary Condition, is necessary for stability of non-periodic ground states against small perturbations of interactions and thermal fluctuations.