A phase transition in block-weighted random maps

TL;DR

This paper studies a phase transition in weighted random planar maps, revealing different geometric behaviors and scaling limits depending on the bias parameter, including Brownian sphere, Brownian tree, and stable tree limits.

Contribution

It identifies a critical bias value where the structure of random planar maps changes, and characterizes the geometric and scaling properties on either side and at the transition.

Findings

For u<u_C, largest block is linear in size, and the map converges to the Brownian sphere.

For u>u_C, largest block is logarithmic, and the map converges to the Brownian tree.

At u=u_C, largest block scales as n^{2/3}, with a stable tree limit.

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 $\Theta(n)$. 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 $\Theta(\log(n))$, 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 $\Theta(n^{2/3})$, the scaling order for distances is $n^{1/3}$, and the scaling limit is the stable tree of parameter $3/2$.