  We study large uniform random maps with one face whose genus grows linearly
with the number of edges, which are a model of discrete hyperbolic geometry. In
previous works, several hyperbolic geometric features have been investigated.
In the present work, we study the number of short cycles in a uniform
unicellular map of high genus, and we show that it converges to a Poisson
distribution. As a corollary, we obtain the law of the systole of uniform
unicellular maps in high genus. We also obtain the asymptotic distribution of
the vertex degrees in such a map.
