布朗球體中的蛇

The snake in the Brownian sphere

February 18, 2025

作者: Omer Angel, Emmanuel Jacob, Brett Kolesnik, Grégory Miermont

cs.AI

摘要

布朗球體是一種隨機度量空間，它與二維球面同胚，並作為多種類型隨機平面圖的普適標度極限而出現。布朗球體的直接構造是通過科里-沃克蘭-謝弗（CVS）雙射的連續類比實現的。CVS雙射將標記樹映射到平面圖，而其連續版本則將帶有布朗標籤的阿爾杜斯連續隨機樹（即布朗蛇）映射到布朗球體。在本研究中，我們通過將布朗蛇構建為布朗球體的可測函數，描述了連續CVS雙射的逆映射。在處理布朗球體的方向時，需要特別注意。

English

The Brownian sphere is a random metric space, homeomorphic to the two-dimensional sphere, which arises as the universal scaling limit of many types of random planar maps. The direct construction of the Brownian sphere is via a continuous analogue of the Cori--Vauquelin--Schaeffer (CVS) bijection. The CVS bijection maps labeled trees to planar maps, and the continuous version maps Aldous' continuum random tree with Brownian labels (the Brownian snake) to the Brownian sphere. In this work, we describe the inverse of the continuous CVS bijection, by constructing the Brownian snake as a measurable function of the Brownian sphere. Special care is needed to work with the orientation of the Brownian sphere.