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

Abstract

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 $(p_n{)}_{n\ge 0}$ 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 ${\mathbf{T}}_n^{\ast }$ 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 ${\mathbf{T}}_n^{\ast }$, in addition to vary along the generations $0\le k\le n-1$, also depend on~$n$.

50 pages, comments welcome. The second version improves the moment condition of the main theorem

References in corpus (7)

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• Ising models on locally tree-like graphs
• Ising models on power-law random graphs
• The critical Ising model on trees, concave recursions and nonlinear capacity
• Extremal inhomogeneous Gibbs states for SOS-models and finite-spin models on trees
• Incoherent boundary conditions and metastates
• Gibbs Measures and Phase Transitions on Sparse Random Graphs