The Blume-Emery-Griffiths model at the FAD and AD interfaces
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866929307738177536 |
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| author | Lima, Paulo C. Mariani, Riccardo Procacci, Aldo Scoppola, Benedetto |
| author_facet | Lima, Paulo C. Mariani, Riccardo Procacci, Aldo Scoppola, Benedetto |
| contents | We analyse the Blume-Emery-Griffiths (BEG) model on the lattice $\Zd$ at the ferromagnetic-antiquadrupolar-disordered (FAD) and antiquadrupolar-disordered (AD) interfaces of parameters. In our analysis of the FAD interface we introduce a Gibbs sampler of the ground states at zero temperature, and we exploit it in two different ways: first, we perform via perfect sampling an empirical evaluation of the spontaneous magnetization at zero temperature, finding a non-zero value in $d=3$ and a vanishing value in $d=2$. Second, using a careful coupling with the Bernoulli site percolation model in $d=2$, we prove rigorously that imposing $+$ boundary conditions, the magnetization in the center of a square box tends to zero in the thermodynamical limit and the two-point correlations decay exponentially. Also, using again a coupling argument, we show that the infinite volume Gibbs measure of the zero-temperature BEG exists and it is unique. In our analysis of the AD interface we restrict ourselves to $d=2$ and, by comparing the BEG model with a Bernoulli site percolation in a matching graph of $\mathbb{Z}^2$, we get a condition for the vanishing of the infinite volume limit magnetization improving, for low temperatures, earlier results obtained via expansion techniques. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_09583 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The Blume-Emery-Griffiths model at the FAD and AD interfaces Lima, Paulo C. Mariani, Riccardo Procacci, Aldo Scoppola, Benedetto Mathematical Physics Probability 82B05 We analyse the Blume-Emery-Griffiths (BEG) model on the lattice $\Zd$ at the ferromagnetic-antiquadrupolar-disordered (FAD) and antiquadrupolar-disordered (AD) interfaces of parameters. In our analysis of the FAD interface we introduce a Gibbs sampler of the ground states at zero temperature, and we exploit it in two different ways: first, we perform via perfect sampling an empirical evaluation of the spontaneous magnetization at zero temperature, finding a non-zero value in $d=3$ and a vanishing value in $d=2$. Second, using a careful coupling with the Bernoulli site percolation model in $d=2$, we prove rigorously that imposing $+$ boundary conditions, the magnetization in the center of a square box tends to zero in the thermodynamical limit and the two-point correlations decay exponentially. Also, using again a coupling argument, we show that the infinite volume Gibbs measure of the zero-temperature BEG exists and it is unique. In our analysis of the AD interface we restrict ourselves to $d=2$ and, by comparing the BEG model with a Bernoulli site percolation in a matching graph of $\mathbb{Z}^2$, we get a condition for the vanishing of the infinite volume limit magnetization improving, for low temperatures, earlier results obtained via expansion techniques. |
| title | The Blume-Emery-Griffiths model at the FAD and AD interfaces |
| topic | Mathematical Physics Probability 82B05 |
| url | https://arxiv.org/abs/2207.09583 |