大偏差局部极限定理与双条件树和映射的极限
Large deviation Local Limit Theorems and limits of biconditioned Trees and Maps
AI总结：
本文建立非递减整数值随机游走的大偏差局部极限，并用于推导双条件 Bienaymé-Galton-Watson 树及随机二部平面映射的标度极限，确认极限为 Brownian map。
摘要：
我们首先为非递减整数值随机游走在时刻 $n$ 位于任意值的概率建立了新的局部极限估计，尤其涵盖了大偏差情形。这使我们能够在各种情形下推导出此类随机游走在受时刻 $n$ 的终值条件约束时的标度极限。我们认为这两者本身都具有独立的研究价值。随后，我们应用这些结果，得到了同时受固定叶子数和顶点数条件约束的 Bienaymé-Galton-Watson 树的 Lukasiewicz 路径的不变性原理，这构成了理解其大尺度几何结构的第一步。最后，我们由此推导出随机二部平面映射在一种新条件下的标度极限定理，该条件同时固定其顶点数、边数和面数。在均匀分布的特殊情形下，我们的结果证实了 Fusy 与 Guitter 关于典型距离增长的预测，并进一步表明在所有情形下，标度极限都是著名的 Brownian map。
英文摘要：
We first establish new local limit estimates for the probability that a nondecreasing integer-valued random walk lies at time $n$ at an arbitrary value, encompassing in particular large deviation regimes. This enables us to derive scaling limits of such random walks conditioned by their terminal value at time $n$ in various regimes. We believe both to be of independent interest. We then apply these results to obtain invariance principles for the Lukasiewicz path of Bienaymé-Galton-Watson trees conditioned on having a fixed number of leaves and of vertices at the same time, which constitutes a first step towards understanding their large scale geometry. We finally deduce from this scaling limit theorems for random bipartite planar maps under a new conditioning by fixing their number of vertices, edges, and faces at the same time. In the particular case of the uniform distribution, our results confirm a prediction of Fusy & Guitter on the growth of the typical distances and show furthermore that in all regimes, the scaling limit is the celebrated Brownian map.
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