An axiomatic characterization of the Brownian map

Miller, Jason; Sheffield, Scott

Journal de l’École polytechnique - Mathématiques, Tome 8 (2021), pp. 609-731.

La carte brownienne est un espace métrique mesuré aléatoire homéomorphe à une sphère, qui est construit en « recollant » les arbres continus décrits respectivement par l’abscisse et l’ordonnée d’un serpent brownien. Nous présentons une construction alternative, reliée au processus d’épluchage ou au cactus brownien, qui produit une surface à partir d’un certain processus de branchement décoré, correspondant à un parcours « en largeur » de la carte brownienne par une exploration.

En utilisant ces idées, nous montrons que la carte brownienne est le seul espace métrique mesuré aléatoire homéomorphe à une sphère possédant certaines propriétés, à savoir l’invariance d’échelle et l’indépendance conditionnelle du côté intérieur et du côté extérieur de certaines « tranches » délimitées par des géodésiques et des bords de boules métriques. Nous formulons aussi une caractérisation en termes du réseau de Lévy produit par une exploration métrique d’un point typique pour la métrique à un autre. Ce résultat est un élément important dans une série d’articles montrant l’équivalence entre la carte brownienne et la sphère en gravité quantique de Liouville de paramètre $\gamma =\sqrt{8/3}$.

The Brownian map is a random sphere-homeomorphic metric measure space obtained by “gluing together” the continuum trees described by the $x$ and $y$ coordinates of the Brownian snake. We present an alternative “breadth-first” construction of the Brownian map, which produces a surface from a certain decorated branching process. It is closely related to the peeling process, the hull process, and the Brownian cactus.

Using these ideas, we prove that the Brownian map is the only random sphere-homeomorphic metric measure space with certain properties: namely, scale invariance and the conditional independence of the inside and outside of certain “slices” bounded by geodesics and metric ball boundaries. We also formulate a characterization in terms of the so-called Lévy net produced by a metric exploration from one measure-typical point to another. This characterization is part of a program for proving the equivalence of the Brownian map and the Liouville quantum gravity sphere with parameter $\gamma =\sqrt{8/3}$.

Classification : 60D05
Keywords: Brownian map, Brownian snake, Brownian tree, Brownian disk, random planar map, Liouville quantum gravity
Mot clés : Carte brownienne, serpent brownien, arbre brownien, disque brownien, carte planaire aléatoire, gravité quantique de Liouville

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