  We study the maximum-average submatrix problem, in which given an $N \times
N$ matrix $J$ one needs to find the $k \times k$ submatrix with the largest
average of entries. We study the problem for random matrices $J$ whose entries
are i.i.d. random variables by mapping it to a variant of the
Sherrington-Kirkpatrick spin-glass model at fixed magnetization. We
characterize analytically the phase diagram of the model as a function of the
submatrix average and the size of the submatrix $k$ in the limit $N\to\infty$.
We consider submatrices of size $k = m N$ with $0 < m < 1$. We find a rich
phase diagram, including dynamical, static one-step replica symmetry breaking
and full-step replica symmetry breaking. In the limit of $m \to 0$, we find a
simpler phase diagram featuring a frozen 1-RSB phase, where the Gibbs measure
is composed of exponentially many pure states each with zero entropy. We
discover an interesting phenomenon, reminiscent of the phenomenology of the
binary perceptron: there exist efficient algorithms that provably work in the
frozen 1-RSB phase.
