Critical-regime covariance-matrix invariants and decompositions

Determine the Jordan normal form, determinant, inverse matrix, eigenspaces, LU decomposition, Cholesky decomposition, and matrix-root decomposition of the asymptotic covariance matrix of a vector of normalized volume power functionals in the critical regime of random Vietoris–Rips complexes, for general admissible parameter sequences.

Background

The paper studies the asymptotic covariance matrix $\Sigma_n$ associated with vectors of normalized volume power functionals of random Vietoris–Rips complexes in the subcritical, critical, and supercritical regimes. The tables in the main-results section classify which algebraic properties have been established in each regime.

For the critical regime, the table marks the Jordan normal form, determinant, inverse matrix, eigenspaces, LU decomposition, Cholesky decomposition, and matrix-root decomposition with $\times$, indicating that these issues remain open. The paper contrasts these gaps with results for rank, definiteness, and determinant positivity except at finitely many parameter values, and with partial eigenvalue bounds under an additional requirement.