Asymptotic decay of coherent-state overlap deficit with maximal-volume eigenstate

Determine the precise asymptotic functional form, as the boundary spin j becomes large, of the fractional deficit D(j) = |⟨Ψ^t_{Γ,{g}}|λ_max⟩⟨λ_max|Ψ^t_{Γ,{g}}⟩ / ⟨Ψ^t_{Γ,{g}}|Ψ^t_{Γ,{g}}⟩ − 1| for the gauge-invariant 4-valent vertex (tetrahedral) configurations with boundary spins j·(1,1,1,1), j·(1,2,1,2), j·(1,3,1,3), and j·(1,4,1,4), thereby deriving the exact decay rate and its functional dependence on j.

Background

The paper studies the eigensystem of the Loop Quantum Gravity volume operator at a vertex and analyzes the overlap between its eigenvectors and gauge-invariant coherent states. For 4-valent vertices corresponding to tetrahedra with boundary spins scaled by j, the authors define a fractional deficit measuring how much the maximal-volume eigenstate saturates the coherent-state norm.

Numerical results suggest an emergent logarithmic-linear behavior in a moderate j window, but computational limits prevent pushing to very large j to conclusively determine the asymptotic form. The authors explicitly state that deriving the precise functional form of the decay rate is left open.