Masur--Veech volumes of quadratic differentials and their asymptotics

Based on the Chen–M"oller–Sauvaget formula, we apply the theory of integrable systems to derive three equations for the generating series of the Masur–Veech volumes $\mathrm{V}\mathrm{o}\mathrm{l}{\mathcal{Q}}_{g,n}$ associated with the principal strata of the moduli spaces of quadratic differentials, and propose refinements of the conjectural formulas given in [12,4] for the large genus asymptotics of $\mathrm{V}\mathrm{o}\mathrm{l}{\mathcal{Q}}_{g,n}$ and of the associated area Siegel–Veech constants.