Square-tiled surfaces of fixed combinatorial type: equidistribution, counting, volumes of the ambient strata

We  prove  that  square-tiled  surfaces (correspondingly pillowcase
covers)  tiled  with  tiny squares sharing a fixed combinatorics of
cylinder  gluings are asymptotically equidistributed in the ambient
stratum  in the moduli space of Abelian (correspondingly quadratic)
differentials.   We  prove  similar  equidistribution  results  for
rational interval exchange transformation.

We  compute  explicitly  the  contribution of square-tiled surfaces
(correspondingly  pillowcase  covers)  having  a  single horizontal
cylinder  to the volume of the corresponding stratum. The resulting
count   is   particularly  simple  and  efficient  in  large  genus
asymptotics. We conjecture that this contribution is asymptotically
of  the  order  $1/d$ where $d$ is the dimension of the stratum and
prove  that  this  conjecture  is  equivalent  to the long-standing
conjecture  on  asymptotics for the volumes of the moduli spaces of
Abelian  differentials.  In certain particular cases the conjecture
was recently proved by D. Chen, M. Moller, and D. Zagier.

Joint work with V.Delecroix, E.Goujard, P.Zograf.