Abstract For general one-matrix models in the large N limit, we introduce the cap amplitude ψ(b) as the expansion coefficient of the 1-form ydx on the spectral curve. We find that the dilaton equation for the discrete volume Ng, n of the moduli space of genus-g Riemann surfaces with n boundaries is interpreted as gluing the cap amplitude along one of the boundaries. In this process, one of the boundaries is capped and the number of boundaries decreases by one. In a similar manner, the genus-g free energy Fg is obtained by gluing the cap amplitude to Ng, 1.
Kazumi Okuyama (Tue,) studied this question.