  Determinants are useful to represent the state of an interacting system of
(effectively) repulsive and independent elements, like fermions in a quantum
system and training samples in a learning problem. A computationally
challenging problem is to compute the sum of powers of principal minors of a
matrix which is relevant to the study of critical behaviors in quantum
fermionic systems and finding a subset of maximally informative training data
for a learning algorithm. Specifically, principal minors of positive square
matrices can be considered as statistical weights of a random point process on
the set of the matrix indices. The probability of each subset of the indices is
in general proportional to a positive power of the determinant of the
associated sub-matrix. We use Gaussian representation of the determinants for
symmetric and positive matrices to estimate the partition function (or free
energy) and the entropy of principal minors within the Bethe approximation. The
results are expected to be asymptotically exact for diagonally dominant
matrices with locally tree-like structures. We consider the Laplacian matrix of
random regular graphs of degree $K=2,3,4$ and exactly characterize the
structure of the relevant minors in a mean-field model of such matrices. No
(finite-temperature) phase transition is observed in this class of diagonally
dominant matrices by increasing the positive power of the principal minors,
which here plays the role of an inverse temperature.
