Ising model on a Galton-Watson tree with a sparse random external field
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| Format: | Preprint |
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2023
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| author | Ventura, Irene Ayuso Berger, Quentin |
| author_facet | Ventura, Irene Ayuso Berger, Quentin |
| contents | We consider the Ising model on a supercritical Galton-Watson tree $\mathbf{T}_n$ of depth $n$ with a sparse random external field, given by a collection of i.i.d. Bernouilli random variables with vanishing parameter $p_n$. This may me viewed as a toy model for the Ising model on a configuration model with a few interfering external vertices carrying a plus spin: the question is to know how many (or how few) interfering vertices are enough to influence the whole graph. Our main result consists in providing a necessary and sufficient condition on the parameters $(p_n)_{n\geq 0}$ for the root of $\mathbf{T}_n$ to remain magnetized in the large $n$ limit. Our model is closely related to the Ising model on a (random) pruned sub-tree $\mathbf{T}_n^*$ with plus boundary condition; one key result is that this pruned tree turns out to be an inhomogeneous, $n$-dependent, Branching Process. We then use standard tools such as tree recursions and non-linear capacities to study the Ising model on this sequence of Galton-Watson trees; one difficulty is that the offspring distributions of $\mathbf{T}_n^*$, in addition to vary along the generations $0\leq k \leq n-1$, also depend on~$n$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_09169 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Ising model on a Galton-Watson tree with a sparse random external field Ventura, Irene Ayuso Berger, Quentin Probability Mathematical Physics 82B20, 60K35, 82B44, 82B26 We consider the Ising model on a supercritical Galton-Watson tree $\mathbf{T}_n$ of depth $n$ with a sparse random external field, given by a collection of i.i.d. Bernouilli random variables with vanishing parameter $p_n$. This may me viewed as a toy model for the Ising model on a configuration model with a few interfering external vertices carrying a plus spin: the question is to know how many (or how few) interfering vertices are enough to influence the whole graph. Our main result consists in providing a necessary and sufficient condition on the parameters $(p_n)_{n\geq 0}$ for the root of $\mathbf{T}_n$ to remain magnetized in the large $n$ limit. Our model is closely related to the Ising model on a (random) pruned sub-tree $\mathbf{T}_n^*$ with plus boundary condition; one key result is that this pruned tree turns out to be an inhomogeneous, $n$-dependent, Branching Process. We then use standard tools such as tree recursions and non-linear capacities to study the Ising model on this sequence of Galton-Watson trees; one difficulty is that the offspring distributions of $\mathbf{T}_n^*$, in addition to vary along the generations $0\leq k \leq n-1$, also depend on~$n$. |
| title | Ising model on a Galton-Watson tree with a sparse random external field |
| topic | Probability Mathematical Physics 82B20, 60K35, 82B44, 82B26 |
| url | https://arxiv.org/abs/2310.09169 |