### Abstract

Consider the classical (2 + 1)-dimensional Solid-On-Solid model above a hard wall on an L×L box of ℤ^{2}. The model describes a crystal surface by assigning a nonnegative integer height η_{x} to each site x in the box and 0 heights to its boundary. The probability of a surface configuration η is proportional to exp(-βH(η)), where β is the inverse-temperature and H(η) sums the absolute values of height differences between neighboring sites. We give a full description of the shape of the SOS surface for low enough temperatures. First we show that with high probability (w.h.p.) the height of almost all sites is concentrated on two levels, H(L) = z(1=4β) logLz and H(L)-1. Moreover, for most values of L the height is concentrated on the single value H(L). Next, we study the ensemble of level lines corresponding to the heights (H(L), H(L)-1,...). We prove that w.h.p. there is a unique macroscopic level line for each height. Furthermore, when taking a diverging sequence of system sizes Lk, the rescaled macroscopic level line at height H(Lk)-n has a limiting shape if the fractional parts of (1=4β) logLk converge to a noncritical value. The scaling limit is an explicit convex subset of the unit square Q and its boundary has a flat component on the boundary of Q. Finally, the highest macroscopic level line has L^{1/3+o(1)} _{k} fluctuations along the flat part of the boundary of its limiting shape.

Original language | English (US) |
---|---|

Pages (from-to) | 991-995 |

Number of pages | 5 |

Journal | Journal of the European Mathematical Society |

Volume | 18 |

Issue number | 5 |

DOIs | |

State | Published - 2016 |

### Fingerprint

### Keywords

- Loop ensembles
- Random surface models
- Scaling limits
- SOS model

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Journal of the European Mathematical Society*,

*18*(5), 991-995. https://doi.org/10.4171/JEMS/606

**Scaling limit and cube-root fluctuations in SOS surfaces above a wall.** / Caputo, Pietro; Lubetzky, Eyal; Martinelli, Fabio; Sly, Allan; Toninelli, Fabio Lucio.

Research output: Contribution to journal › Article

*Journal of the European Mathematical Society*, vol. 18, no. 5, pp. 991-995. https://doi.org/10.4171/JEMS/606

}

TY - JOUR

T1 - Scaling limit and cube-root fluctuations in SOS surfaces above a wall

AU - Caputo, Pietro

AU - Lubetzky, Eyal

AU - Martinelli, Fabio

AU - Sly, Allan

AU - Toninelli, Fabio Lucio

PY - 2016

Y1 - 2016

N2 - Consider the classical (2 + 1)-dimensional Solid-On-Solid model above a hard wall on an L×L box of ℤ2. The model describes a crystal surface by assigning a nonnegative integer height ηx to each site x in the box and 0 heights to its boundary. The probability of a surface configuration η is proportional to exp(-βH(η)), where β is the inverse-temperature and H(η) sums the absolute values of height differences between neighboring sites. We give a full description of the shape of the SOS surface for low enough temperatures. First we show that with high probability (w.h.p.) the height of almost all sites is concentrated on two levels, H(L) = z(1=4β) logLz and H(L)-1. Moreover, for most values of L the height is concentrated on the single value H(L). Next, we study the ensemble of level lines corresponding to the heights (H(L), H(L)-1,...). We prove that w.h.p. there is a unique macroscopic level line for each height. Furthermore, when taking a diverging sequence of system sizes Lk, the rescaled macroscopic level line at height H(Lk)-n has a limiting shape if the fractional parts of (1=4β) logLk converge to a noncritical value. The scaling limit is an explicit convex subset of the unit square Q and its boundary has a flat component on the boundary of Q. Finally, the highest macroscopic level line has L1/3+o(1) k fluctuations along the flat part of the boundary of its limiting shape.

AB - Consider the classical (2 + 1)-dimensional Solid-On-Solid model above a hard wall on an L×L box of ℤ2. The model describes a crystal surface by assigning a nonnegative integer height ηx to each site x in the box and 0 heights to its boundary. The probability of a surface configuration η is proportional to exp(-βH(η)), where β is the inverse-temperature and H(η) sums the absolute values of height differences between neighboring sites. We give a full description of the shape of the SOS surface for low enough temperatures. First we show that with high probability (w.h.p.) the height of almost all sites is concentrated on two levels, H(L) = z(1=4β) logLz and H(L)-1. Moreover, for most values of L the height is concentrated on the single value H(L). Next, we study the ensemble of level lines corresponding to the heights (H(L), H(L)-1,...). We prove that w.h.p. there is a unique macroscopic level line for each height. Furthermore, when taking a diverging sequence of system sizes Lk, the rescaled macroscopic level line at height H(Lk)-n has a limiting shape if the fractional parts of (1=4β) logLk converge to a noncritical value. The scaling limit is an explicit convex subset of the unit square Q and its boundary has a flat component on the boundary of Q. Finally, the highest macroscopic level line has L1/3+o(1) k fluctuations along the flat part of the boundary of its limiting shape.

KW - Loop ensembles

KW - Random surface models

KW - Scaling limits

KW - SOS model

UR - http://www.scopus.com/inward/record.url?scp=84963571335&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84963571335&partnerID=8YFLogxK

U2 - 10.4171/JEMS/606

DO - 10.4171/JEMS/606

M3 - Article

AN - SCOPUS:84963571335

VL - 18

SP - 991

EP - 995

JO - Journal of the European Mathematical Society

JF - Journal of the European Mathematical Society

SN - 1435-9855

IS - 5

ER -