On the growth of random planar maps with a prescribed degree sequence

For non-negative integers $(d_n(k){)}_{k\ge 1}$ such that $\sum_{k \ge 1} d_n(k) = n$,wesampleabipartiteplanarmapwith$n$ faces uniformly at random amongst those which have $d_n(k)$ faces of degree $2k$ for every $k \ge 1$andwestudyitsasymptoticbehaviouras$n \to \infty$. We prove that the diameter of such maps grow like ${\sigma }_n^{1\mathrm{/}2}$, where $\sigma_n^2 = \sum_{k \ge 1} k (k-1) d_n(k)$ is a global variance term. More precisely, we prove that the vertex-set of these maps equipped with the graph distance divided by ${\sigma }_n^{1\mathrm{/}2}$ and the uniform probability measure always admits subsequential limits in the Gromov-Hausdorff-Prokhorov topology. Our proof relies on a bijection with random labelled trees; we are able to prove that the label process is always tight when suitably rescaled, even if the underlying tree is not tight for the Gromov-Hausdorff topology. We also rely on a new spinal decomposition which is of independent interest. Finally this paper also serves as a toolbox for a companion paper in which we discuss more precisely Brownian limits of such maps.