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arXiv 2402.04827math.PRmath-phmath.MP
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循环O(n)四边形的体积缩放极限

The scaling limit of the volume of loop O(n) quadrangulations

Fudan University（复旦大学）；University of Vienna（维也纳大学）

Élie Aïdékon, William Da Silva, Xingjian Hu

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67 pages, 10 figures
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AI总结：

研究了循环O(n)四边形体积的缩放极限，揭示了其与临界分支随机游走的关系，并给出了反指数分布的结论。

摘要：

我们研究了具有长度为2p边界的刚性循环O(n)四边形的体积，在非通用临界情况下。我们证明，当半周长p趋于无穷大时，体积在分布上趋于一个显式的随机变量。这个极限随机变量通过Chen、Curien和Maillard的乘法级数（arXiv:1702.06916）或替代地（在稀疏情况下）通过Ang和Gwynne确定的单位边界γ-量子盘的面积分布（arXiv:1903.09120）来描述，对于合适的γ。我们的论证通过将地图划分为几个区域进行，其中我们排除了坏区域的贡献，仅留下可处理的地图部分。这种分类的一个关键可观测变量是一个马尔可夫链，它在地图中围绕一个大小偏向的顶点进行嵌套循环探索，明确揭示了离散乘法级数的脊柱结构。我们强调，我们的技术使我们能够包括边界情况n=2，我们对其进行了严格定义，其中嵌套级数结构是临界分支随机游走。在这种情况下，缩放极限由导数鞅的极限给出，并呈反指数分布，这回答了arXiv:2005.06372v2中的猜想。

英文摘要：

We study the volume of rigid loop-$O(n)$ quadrangulations with a boundary of length $2p$ in the non-generic critical regime. We prove that, as the half-perimeter $p$ goes to infinity, the volume scales in distribution to an explicit random variable. This limiting random variable is described in terms of the multiplicative cascades of Chen, Curien and Maillard arXiv:1702.06916, or alternatively (in the dilute case) as the law of the area of a unit-boundary $γ$-quantum disc, as determined by Ang and Gwynne arXiv:1903.09120, for suitable $γ$. Our arguments go through a classification of the map into several regions, where we rule out the contribution of bad regions to be left with a tractable portion of the map. One key observable for this classification is a Markov chain which explores the nested loops around a size-biased vertex pick in the map, making explicit the spinal structure of the discrete multiplicative cascade. We stress that our techniques enable us to include the boundary case $n=2$, that we define rigorously, and where the nested cascade structure is that of a critical branching random walk. In that case the scaling limit is given by the limit of the derivative martingale and is inverse-exponentially distributed, which answers a conjecture of arXiv:2005.06372v2.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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