  We consider testing for the parameters of Ferromagnetic Ising models. While
testing for the presence of possibly sparse magnetizations, we provide a
general lower bound of minimax separation rates which yields sharp results in
high temperature regimes. Our matching upper bounds are adaptive over both
underlying dependence graph and temperature parameter. Moreover our results
include the nearest neighbor model on lattices, the sparse Erd\"{o}s-R\'{e}nyi
random graphs, and regular rooted trees -- right up to the critical parameter
in the high temperature regime. We also provide parallel results for the entire
low temperature regime in nearest neighbor model on lattices -- however in the
plus boundary pure phase. Our results for the nearest neighbor model crucially
depends on finite volume analogues of correlation decay property for both high
and low temperature regimes -- the derivation of which borrows crucial ideas
from FK-percolation theory and might be of independent interest. Finally, we
also derive lower bounds for estimation and testing rates in two parameter
Ising models -- which turn out to be optimal according to several recent
results in this area.
