  We investigate the structure of large uniform random maps with $n$ edges,
$\mathrm{f}_n$ faces, and with genus $\mathrm{g}_n$ in the so-called sparse
case, where the ratio between the number vertices and edges tends to $1$. We
focus on two regimes: the planar case $(\mathrm{f}_n, 2\mathrm{g}_n) =
(\mathrm{s}_n, 0)$ and the unicellular case with moderate genus $(\mathrm{f}_n,
2 \mathrm{g}_n) = (1, \mathrm{s}_n-1)$, both when $1 \ll \mathrm{s}_n \ll n$.
Albeit different at first sight, these two models can be treated in a unified
way using a probabilistic version of the classical core-kernel decomposition.
In particular, we show that the number of edges of the core of such maps,
obtained by iteratively removing degree $1$ vertices, is concentrated around
$\sqrt{n \mathrm{s}_{n}}$. Further, their kernel, obtained by contracting the
vertices of the core with degree $2$, is such that the sum of the degree of its
vertices exceeds that of a trivalent map by a term of order
$\sqrt{\mathrm{s}_{n}^{3}/n}$; in particular they are trivalent with high
probability when $\mathrm{s}_{n} \ll n^{1/3}$. This enables us to identify a
mesoscopic scale $\sqrt{n/\mathrm{s}_n}$ at which the scaling limits of these
random maps can be seen as the local limit of their kernels, which is the dual
of the UIPT in the planar case and the infinite three-regular tree in the
unicellular case, where each edge is replaced by an independent (biased)
Brownian tree with two marked points.
