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

We consider the Ising model on a supercritical Galton–Watson tree Tn of depth n with a sparse random external field, given by a collection of i.i.d. Bernoulli random variables with vanishing parameter pn. 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 (pn)n⩾0 for the root of Tn to remain magnetized in the large n limit. Our model is closely related to the Ising model on a (random) pruned sub-tree Tn∗ 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 Tn∗, in addition to vary along the generations 0⩽k⩽n−1, also depend on n.