  We study the masses of the two cells in a Vorono\"i tessellation of the
Brownian surface of genus $g\geq 0$ centered on two uniform random points.
Making use of classical bijections and asymptotic estimates for maps of fixed
genus, we relate the second moment of these random variables to the
Painlev\'e-I equation satisfied by the double scaling limit of the one-matrix
model, or equivalently to the "$t_g$-recurrence" satisfied by the constants
$t_g$ driving the asymptotic number of maps of genus $g\geq0$. This raises the
question of giving an independent probabilistic or combinatorial derivation of
this second moment, which would then lead to new proof of the $t_g$-recurrence.
  More generally we conjecture that for any $g\geq 0$ and $k\geq 2$, the masses
of the cells in a Vorono\"i tessellation of the genus-$g$ Brownian surface by
$k$ uniform points follows a Dirichlet$(1,1,\dots,1)$ distribution.
