Baxter置换通过合并行走过程的标度和局部极限
Scaling and local limits of Baxter permutations through coalescent-walk processes
摘要：
众所周知，巴克斯特排列、平面双极取向和非负象限中的一个特定的行走族通过几个双射相互关联。我们引入了一个新的家庭离散对象，称为聚结行走过程，这是我们的结果的基础。我们将这些新的对象与其他先前提到的家庭引入一些新的双射。我们证明联合Benjamini-Schramm收敛（无论是在退火和淬火的意义上）的四个家庭中的统一对象。此外，我们明确地构建了一个新的分形随机测度的单位平方，称为合并巴克斯特置换，我们表明，它是标度极限（在置换意义上）的均匀巴克斯特置换。为了证明后一个结果，我们研究了相关的随机合并行走过程的标度极限。我们表明，他们收敛于法律的连续随机聚结行走过程编码的扰动版本的田中随机微分方程。这一结果与Gwynne，Holden，Sun（2016）关于平面双极三角剖分的缩放限制（在Peanosphere拓扑中）的结果有联系（将在未来的项目中探索）。我们进一步证明了一些结果，涉及的限制对象的四个家庭彼此，无论是在本地和缩放限制的情况下。
英文摘要：
Baxter permutations, plane bipolar orientations, and a specific family of walks in the non-negative quadrant are well-known to be related to each other through several bijections. We introduce a further new family of discrete objects, called coalescent-walk processes, that are fundamental for our results. We relate these new objects with the other previously mentioned families introducing some new bijections. We prove joint Benjamini--Schramm convergence (both in the annealed and quenched sense) for uniform objects in the four families. Furthermore, we explicitly construct a new fractal random measure of the unit square, called the coalescent Baxter permuton and we show that it is the scaling limit (in the permuton sense) of uniform Baxter permutations. To prove the latter result, we study the scaling limit of the associated random coalescent-walk processes. We show that they converge in law to a continuous random coalescent-walk process encoded by a perturbed version of the Tanaka stochastic differential equation. This result has connections (to be explored in future projects) with the results of Gwynne, Holden, Sun (2016) on scaling limits (in the Peanosphere topology) of plane bipolar triangulations. We further prove some results that relate the limiting objects of the four families to each other, both in the local and scaling limit case.
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