Entropic repulsion of 3D Ising interfaces conditioned to stay above a floor
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| Format: | Preprint |
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2021
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| author | Gheissari, Reza Lubetzky, Eyal |
| author_facet | Gheissari, Reza Lubetzky, Eyal |
| contents | We study the interface of the Ising model in a box of side-length $n$ in $\mathbb Z^3$ at low temperature $1/β$ under Dobrushin's boundary conditions, conditioned to stay in a half-space above height $h$ (a hard floor). Without this conditioning, Dobrushin showed in 1972 that typically most of the interface is flat at height $0$. With the floor, for small $h$, the model is expected to exhibit {\it entropic repulsion}, where the typical height of the interface lifts off of $0$. Detailed understanding of the SOS model -- a more tractable height function approximation of 3D Ising -- due to Caputo et al., suggests that there is a single integer value $-h_n^* \sim -c\log n$ of the floor height, delineating the transition between rigidity at height $0$ and entropic repulsion.
We identify an explicit $h_n^*=( c_\star+o(1))\log n$ such that, for the typical Ising interface above a hard floor at $h$, all but an $ε(β)$-fraction of the sites are propelled to be above height $0$ if $h < h_n^*-1$, whereas all but an $ε(β)$-fraction of the sites remain at height $0$ if $h\geq h_n^*$. Further, $c_\star$ is such that the typical height of the unconditional maximum is $(2c_\star + o(1))\log n$; this confirms scaling predictions from the SOS approximation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2112_05133 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Entropic repulsion of 3D Ising interfaces conditioned to stay above a floor Gheissari, Reza Lubetzky, Eyal Probability Mathematical Physics 60K35, 82B20, 82B24, 82B41 We study the interface of the Ising model in a box of side-length $n$ in $\mathbb Z^3$ at low temperature $1/β$ under Dobrushin's boundary conditions, conditioned to stay in a half-space above height $h$ (a hard floor). Without this conditioning, Dobrushin showed in 1972 that typically most of the interface is flat at height $0$. With the floor, for small $h$, the model is expected to exhibit {\it entropic repulsion}, where the typical height of the interface lifts off of $0$. Detailed understanding of the SOS model -- a more tractable height function approximation of 3D Ising -- due to Caputo et al., suggests that there is a single integer value $-h_n^* \sim -c\log n$ of the floor height, delineating the transition between rigidity at height $0$ and entropic repulsion. We identify an explicit $h_n^*=( c_\star+o(1))\log n$ such that, for the typical Ising interface above a hard floor at $h$, all but an $ε(β)$-fraction of the sites are propelled to be above height $0$ if $h < h_n^*-1$, whereas all but an $ε(β)$-fraction of the sites remain at height $0$ if $h\geq h_n^*$. Further, $c_\star$ is such that the typical height of the unconditional maximum is $(2c_\star + o(1))\log n$; this confirms scaling predictions from the SOS approximation. |
| title | Entropic repulsion of 3D Ising interfaces conditioned to stay above a floor |
| topic | Probability Mathematical Physics 60K35, 82B20, 82B24, 82B41 |
| url | https://arxiv.org/abs/2112.05133 |