  We study the problem of testing and recovering the hidden $k$-clique
Ferromagnetic correlation in the planted Random Field Curie-Weiss model (a.k.a.
the pRFCW model). The pRFCW model is a random effect Ising model that exhibits
richer phase diagrams both statistically and physically than the standard
Curie-Weiss model. Using an alternative characterization of parameter regimes
as 'temperatures' and the mean values as 'outer magnetic fields,' we establish
the minimax optimal detection rates and recovery rates. The results consist of
$7$ distinctive phases for testing and $3$ phases for exact recovery. Our
results also imply that the randomness of the outer magnetic field contributes
to countable possible convergence rates, which are not observed in the fixed
field model. As a byproduct of the proof techniques, we provide two new
mathematical results: (1) A family of tail bounds for the average magnetization
of the Random Field Curie-Weiss model (a.k.a. the RFCW model) across all
temperatures and arbitrary outer fields. (2) A sharp estimate of the
information divergence between RFCW models. These play pivotal roles in
establishing the major theoretical results in this paper. Additionally, we show
that the mathematical structure involved in the pRFCW hidden clique inference
problem resembles a 'sparse PCA-like' problem for discrete data. The richer
statistical phases than the long-studied Gaussian counterpart shed new light on
the theoretical insight of sparse PCA for discrete data.
