A phase transition in block-weighted random maps

Abstract

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 $u_C=9∕5$. 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 $n^{1∕4}$, 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 $n^{1∕2}$, and the scaling limit is the Brownian tree. Finally, for $u=u_C$, the largest block is of size $\mathrm{\Theta }(n^{2∕3})$, the scaling order for distances is $n^{1∕3}$, and the scaling limit is the stable tree of parameter $3∕2$.