Closed analytic form for hopping-expansion sums

Derive an analytic expression for the hopping-expansion sums defining the matrices $M_{ij}$, $N_l$, and $\overline{N}_l$ for the entanglement entropy of the discretized scalar field.

Background

The hopping expansion represents the projected covariance matrix and its associated symplectic eigenvalues through recursively generated path sums. These sums are evaluated numerically in the paper, with fixed-order and resummed truncations used to approximate the entropy scaling function.

An analytic expression for the sums would provide a more systematic understanding of convergence and could improve or replace the numerical truncation and resummation procedures used to obtain the large-effective-mass entropy behavior.

References

Since we were unable to find an analytic expression for mij and nlnbl, we evaluated the truncated sums numerically.

— Entropy, area, and the choice of regulator during gravitational collapse  (2609.20663 - Guenther et al., 17 Sep 2026) in Appendix, Section “Hopping expansion”

Combining it with the infinite volume effective mass term from sassyinf, one may conjecture a continuum entanglement entropy for finite effective mass and volume

— Entropy, area, and the choice of regulator during gravitational collapse  (2609.20663 - Guenther et al., 17 Sep 2026) in Appendix, Section “Asymptotic scaling from the harmonic oscillator chain approximation”