Tropicalizing Principal Minors of Positive Definite Matrices

We study the tropicalization of principal minors of positive definite matrices over a real valued field. This tropicalization forms a subset of M-concave functions on the discrete n-dimensional cube. We show that it coincides with a linear slice of a larger tropical Grassmannian, and also with the portion of the affine tropical Flag variety inside the submodular cone.  By lifting tropical inequalities, we uncover new polynomial inequalities on the principal minors, and more generally, on coefficients of Lorentzian polynomials.