Article Dans Une Revue Electronic Journal of Probability Année : 2024

Ising model on a Galton-Watson tree with a sparse random external field

Résumé

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~$n$.

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Dates et versions

hal-04253083 , version 1 (21-10-2023)
hal-04253083 , version 2 (14-08-2025)

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Irene Ayuso Ventura, Quentin Berger. Ising model on a Galton-Watson tree with a sparse random external field. Electronic Journal of Probability, 2024, 29, pp.1-43. ⟨10.1214/24-EJP1232⟩. ⟨hal-04253083v2⟩
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