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

Pietro Caputo; Eyal Lubetzky; Fabio Martinelli; Allan Sly; Fabio Lucio Toninelli

Journal of the European Mathematical Society (2016)

- Volume: 018, Issue: 5, page 931-995
- ISSN: 1435-9855

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topCaputo, Pietro, et al. "Scaling limit and cube-root fluctuations in SOS surfaces above a wall." Journal of the European Mathematical Society 018.5 (2016): 931-995. <http://eudml.org/doc/277798>.

@article{Caputo2016,

abstract = {Consider the classical $(2+1)$-dimensional Solid-On-Solid model above a hard wall on an $L \times L$ box of $\mathbb \{Z\}^2$. The model describes a crystal surface by assigning a non-negative integer height $\eta _x$ to each site $x$ in the box and 0 heights to its boundary. The probability of a surface configuration $\eta $ is proportional to $\mathrm \{exp\}(-\beta \mathcal \{H\}(\eta ))$, where $\beta $ is the inverse-temperature and $\mathcal \{H\}(\eta )$ 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) = \lfloor (1/4\beta )\mathrm \{log\} L \rfloor $ 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 $(H(L),H(L)-1,\ldots )$. 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 $L_k$, the rescaled macroscopic level line at height $H(L_k)-n$ has a limiting shape if the fractional parts of $(1/4\beta )\mathrm \{log\} L_k$ 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 $L_k^\{1/3+o(1)\}$ fluctuations along the flat part of the boundary of its limiting shape.},

author = {Caputo, Pietro, Lubetzky, Eyal, Martinelli, Fabio, Sly, Allan, Toninelli, Fabio Lucio},

journal = {Journal of the European Mathematical Society},

keywords = {SOS model; scaling limits; loop ensembles; random surface models; SOS model; scaling limits; loop ensembles; random surface models},

language = {eng},

number = {5},

pages = {931-995},

publisher = {European Mathematical Society Publishing House},

title = {Scaling limit and cube-root fluctuations in SOS surfaces above a wall},

url = {http://eudml.org/doc/277798},

volume = {018},

year = {2016},

}

TY - JOUR

AU - Caputo, Pietro

AU - Lubetzky, Eyal

AU - Martinelli, Fabio

AU - Sly, Allan

AU - Toninelli, Fabio Lucio

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

JO - Journal of the European Mathematical Society

PY - 2016

PB - European Mathematical Society Publishing House

VL - 018

IS - 5

SP - 931

EP - 995

AB - Consider the classical $(2+1)$-dimensional Solid-On-Solid model above a hard wall on an $L \times L$ box of $\mathbb {Z}^2$. The model describes a crystal surface by assigning a non-negative integer height $\eta _x$ to each site $x$ in the box and 0 heights to its boundary. The probability of a surface configuration $\eta $ is proportional to $\mathrm {exp}(-\beta \mathcal {H}(\eta ))$, where $\beta $ is the inverse-temperature and $\mathcal {H}(\eta )$ 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) = \lfloor (1/4\beta )\mathrm {log} L \rfloor $ 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 $(H(L),H(L)-1,\ldots )$. 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 $L_k$, the rescaled macroscopic level line at height $H(L_k)-n$ has a limiting shape if the fractional parts of $(1/4\beta )\mathrm {log} L_k$ 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 $L_k^{1/3+o(1)}$ fluctuations along the flat part of the boundary of its limiting shape.

LA - eng

KW - SOS model; scaling limits; loop ensembles; random surface models; SOS model; scaling limits; loop ensembles; random surface models

UR - http://eudml.org/doc/277798

ER -

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