  In this paper, we study the effect of dependence on detecting sparse signals.
In particular, we focus on global testing against sparse alternatives for the
means of binary outcomes following an Ising model, and establish how the
interplay between the strength and sparsity of a signal determines its
detectability under various notions of dependence. The profound impact of
dependence is best illustrated under the Curie-Weiss model where we observe the
effect of a "thermodynamic" phase transition. In particular, the critical state
exhibits a subtle "blessing of dependence" phenomenon in that one can detect
much weaker signals at criticality than otherwise. Furthermore, we develop a
testing procedure that is broadly applicable to account for dependence and show
that it is asymptotically minimax optimal under fairly general regularity
conditions.
