A phase transition in block-weighted random maps

We consider the model of random planar maps of size $n$ biased by a weight $u>0$ per $2$-connected block, and the closely related model of random planar quadrangulations of size $n$ biased by a weight $u>0$ per simple component. We exhibit a phase transition at the critical value $u_C=9\mathrm{/}5$. If $u<u_C$, a condensation phenomenon occurs: the largest block is of size $\mathrm{\Theta }(n)$. Moreover, for quadrangulations we show that the diameter is of order $n^{1\mathrm{/}4}$, and the scaling limit is the Brownian sphere. When $u>u_C$, the largest block is of size $\mathrm{\Theta }(\log (n))$, the scaling order for distances is $n^{1\mathrm{/}2}$, and the scaling limit is the Brownian tree. Finally, for $u=u_C$, the largest block is of size $\mathrm{\Theta }(n^{2\mathrm{/}3})$, the scaling order for distances is $n^{1\mathrm{/}3}$, and the scaling limit is the stable tree of parameter $3\mathrm{/}2$.