  We study uniformly random maps with a single face, genus $g$, and size $n$,
as $n,g\rightarrow \infty$ with $g = o(n)$, in continuation of several previous
works on the geometric properties of "high genus maps". We calculate the number
of short simple cycles, and we show convergence of their lengths (after a
well-chosen rescaling of the graph distance) to a Poisson process, which
happens to be exactly the same as the limit law obtained by Mirzakhani and
Petri (2019) when they studied simple closed geodesics on random hyperbolic
surfaces under the Weil-Petersson measure as $g\rightarrow \infty$. This leads
us to conjecture that these two models are somehow "the same" in the limit,
which would allow to translate problems on hyperbolic surfaces in terms of
random trees, thanks to a powerful bijection of Chapuy, F\'eray and Fusy
(2013).
