Deconstructing the Gel'fand-Yaglom method and vacuum energy from a theory space

The discrete Gel'fand-Yaglom theorem was reviewed and studied by Dowker (J. Phys. A45 (2012) 215203), several years ago. In the present paper, we generalize the discrete Gel'fand-Yaglom method to obtain the determinants of mass matrices which appear current works in particle physics, such as dimensional deconstruction and clockwork theory. Using the results, we show the expressions for vacuum energies in such various models.