块加权随机地图中的相变
A phase transition in block-weighted random maps
AI总结：
本文研究块加权随机地图和四边形剖分的相变，在临界值 $u_C=9/5$ 处，最大块大小和缩放极限从凝聚态（布朗球）过渡到对数态（布朗树），临界点呈现稳定树极限。
摘要：
我们考虑大小为 $n$ 的随机平面地图模型，该模型按每个 $2$-连通块权重 $u>0$ 偏置，以及与之密切相关的随机平面四边形剖分模型，该模型按每个简单分量权重 $u>0$ 偏置。我们在临界值 $u_C=9/5$ 处展示了一个相变。若 $u<u_C$，则发生凝聚现象：最大块的大小为 $\Theta(n)$。此外，对于四边形剖分，我们证明直径的数量级为 $n^{1/4}$，缩放极限为布朗球。当 $u > u_C$ 时，最大块的大小为 $\Theta(\log(n))$，距离的缩放阶为 $n^{1/2}$，缩放极限为布朗树。最后，当 $u=u_C$ 时，最大块的大小为 $\Theta(n^{2/3})$，距离的缩放阶为 $n^{1/3}$，缩放极限为参数 $3/2$ 的稳定树。
英文摘要：
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 $Θ(n)$. 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 $Θ(\log(n))$, 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 $Θ(n^{2/3})$, the scaling order for distances is $n^{1/3}$, and the scaling limit is the stable tree of parameter $3/2$.
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