  We study the geometry of infinite random Boltzmann planar maps having weight
of polynomial decay of order $k^{-2}$ for each vertex of degree $k$. These
correspond to the dual of the discrete "stable maps" of Le Gall and Miermont
[Scaling limits of random planar maps with large faces, Ann. Probab. 39, 1
(2011), 1-69] studied in [Budd & Curien, Geometry of infinite planar maps with
high degrees, Electron. J. Probab. (to appear)] related to a symmetric Cauchy
process, or alternatively to the maps obtained after taking the gasket of a
critical $O(2)$-loop model on a random planar map. We show that these maps have
a striking and uncommon geometry. In particular we prove that the volume of the
ball of radius $r$ for the graph distance has an intermediate rate of growth
and scales as $\mathrm{e}^{\sqrt{r}}$. We also perform first passage
percolation with exponential edge-weights and show that the volume growth for
the fpp-distance scales as $\mathrm{e}^{r}$. Finally we consider site
percolation on these lattices: although percolation occurs only at $p=1$, we
identify a phase transition at $p=1/2$ for the length of interfaces. On the way
we also prove new estimates on random walks attracted to an asymmetric Cauchy
process.

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