  In the 80's H. Masur and W. Veech defined two numerical invariants of strata
of abelian differentials: the volume and the Siegel-Veech constant. Based on
numerical experiments, A. Eskin and A. Zorich proposed a series of conjectures
for the large genus asymptotics of these invariants. By a careful analysis of
the asymptotic behavior of quasi-modular forms, D. Chen, M. Moeller, and D.
Zagier proved this conjecture for strata of differentials with simple zeros.
Here, we prove that the conjecture holds for the other extreme case, i.e. for
strata of differentials with a unique zero. Our main ingredient is the
expression of the numerical invariants of these strata in terms of Hodge
integrals on moduli spaces of curves.
