Columnar order in random packings of $2\times2$ squares on the square lattice
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2022
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866918343317913600 |
|---|---|
| author | Hadas, Daniel Peled, Ron |
| author_facet | Hadas, Daniel Peled, Ron |
| contents | We study random packings of $2\times2$ squares with centers on the square lattice $\mathbb{Z}^{2}$, in which the probability of a packing is proportional to $λ$ to the number of squares. We prove that for large $λ$, typical packings exhibit columnar order, in which either the centers of most tiles agree on the parity of their $x$-coordinate or the centers of most tiles agree on the parity of their $y$-coordinate. This manifests in the existence of four extremal and periodic Gibbs measures in which the rotational symmetry of the lattice is broken while the translational symmetry is only broken along a single axis. We further quantify the decay of correlations in these measures, obtaining a slow rate of exponential decay in the direction of preserved translational symmetry and a fast rate in the direction of broken translational symmetry. Lastly, we prove that every periodic Gibbs measure is a mixture of these four measures.
Additionally, our proof introduces an apparently novel extension of the chessboard estimate, from finite-volume torus measures to all infinite-volume periodic Gibbs measures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_01276 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Columnar order in random packings of $2\times2$ squares on the square lattice Hadas, Daniel Peled, Ron Mathematical Physics Statistical Mechanics Combinatorics Probability We study random packings of $2\times2$ squares with centers on the square lattice $\mathbb{Z}^{2}$, in which the probability of a packing is proportional to $λ$ to the number of squares. We prove that for large $λ$, typical packings exhibit columnar order, in which either the centers of most tiles agree on the parity of their $x$-coordinate or the centers of most tiles agree on the parity of their $y$-coordinate. This manifests in the existence of four extremal and periodic Gibbs measures in which the rotational symmetry of the lattice is broken while the translational symmetry is only broken along a single axis. We further quantify the decay of correlations in these measures, obtaining a slow rate of exponential decay in the direction of preserved translational symmetry and a fast rate in the direction of broken translational symmetry. Lastly, we prove that every periodic Gibbs measure is a mixture of these four measures. Additionally, our proof introduces an apparently novel extension of the chessboard estimate, from finite-volume torus measures to all infinite-volume periodic Gibbs measures. |
| title | Columnar order in random packings of $2\times2$ squares on the square lattice |
| topic | Mathematical Physics Statistical Mechanics Combinatorics Probability |
| url | https://arxiv.org/abs/2206.01276 |