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We prove combinatorially the explicit relation between genus filtrated$s$-loop means of the Gaussian matrix model and terms of the genus expansion ofthe Kontsevich--Penner matrix model (KPMM). The latter is the generatingfunction for volumes of discretized (open) moduli spaces$M_{g,s}^{\mathrm{disc}}$ given by $N_{g,s}(P_1,\dots,P_s)$ for$(P_1,\dots,P_s)\in{\mathbb Z}_+^s$. This generating function therefore enjoysthe topological recursion, and we prove that it is simultaneously thegenerating function for ancestor invariants of a cohomological field theorythus enjoying the Givental decomposition. We use another Givental-typedecomposition obtained for this model by the second authors in 1995 in terms ofspecial times related to the discretisation of moduli spaces thus representingits asymptotic expansion terms (and therefore those of the Gaussian means) asfinite sums over graphs weighted by lower-order monomials in times thus givinganother proof of (quasi)polynomiality of the discrete volumes. As anapplication, we find the coefficients in the first subleading order for${\mathcal M}_{g,1}$ in two ways: using the refined Harer--Zagier recursion andby exploiting the above Givental-type transformation. We put forward theconjecture that the above graph expansions can be used for probing thereduction structure of the Delgne--Mumford compactification $\overline{\mathcalM}_{g,s}$ of moduli spaces of punctured Riemann surfaces.

Models of discretized moduli spaces, cohomological field theories, and Gaussian means