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.

翻译：我们研究最大平均子矩阵问题，在该问题中，给定一个$N \times N$矩阵$J$，需要找到具有最大条目平均值的$k \times k$子矩阵。通过将该问题映射到固定磁化强度下的Sherrington-Kirkpatrick自旋玻璃模型的一个变体，我们研究了条目为独立同分布随机变量的随机矩阵$J$的情况。我们解析地描述了该模型在$N\to\infty$极限下，相图作为子矩阵平均值和子矩阵大小$k$的函数的特征。我们考虑大小为$k = m N$且$0 < m < 1$的子矩阵。我们发现了一个丰富的相图，包括动力学、静态一步复制对称破缺以及全步复制对称破缺。在$m \to 0$的极限下，我们找到了一个更简单的相图，其中包含一个冻结的1-RSB相，此时吉布斯测度由指数多个熵为零的纯态组成。