圆盘的随机稳定叠层
Random stable laminations of the disk
摘要：
我们研究多边形的大随机剖分。我们考虑随机剖分的正多边形的$n$边，这是根据玻尔兹曼权重选择的吸引区域的稳定的指数律$θ\in(1,2]$。当$n$趋于无穷大时，我们证明了这些随机剖分在分布上收敛于一个随机紧集，称为随机稳定层。如果是$θ=2$，我们恢复Aldous的布朗三角剖分。但是，如果为$θ\in(1,2)$，则大面仍在极限范围内，并且会出现不同的随机紧致集。我们证明了随机稳定分层可以用连续时间高度函数编码，该函数与指数为$θ$的严格稳定谱正Lévy过程的归一化偏移相关联.使用这种编码，我们建立了稳定的随机层的豪斯多夫维数几乎肯定是$2-1/θ$。
英文摘要：
We study large random dissections of polygons. We consider random dissections of a regular polygon with $n$ sides, which are chosen according to Boltzmann weights in the domain of attraction of a stable law of index $θ\in(1,2]$. As $n$ goes to infinity, we prove that these random dissections converge in distribution toward a random compact set, called the random stable lamination. If $θ=2$, we recover Aldous' Brownian triangulation. However, if $θ\in(1,2)$, large faces remain in the limit and a different random compact set appears. We show that the random stable lamination can be coded by the continuous-time height function associated to the normalized excursion of a strictly stable spectrally positive Lévy process of index $θ$. Using this coding, we establish that the Hausdorff dimension of the stable random lamination is almost surely $2-1/θ$.
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