Globally centered discrete snakes

The lineage process in Galton--Watson trees and globally centered discrete snakes

We consider branching random walks built on Galton--Watson trees with offspring distribution having a bounded support, conditioned to have $n$ nodes, and their rescaled convergences to the Brownian snake. We exhibit a notion of ``globally centered discrete snake'' that extends the usual settings in which the displacements are supposed centered. We show that under some additional moment conditions, when $n$ goes to $+\infty$, ``globally centered discrete snakes'' converge to the Brownian snake. The proof relies on a precise study of the lineage of the nodes in a Galton--Watson tree conditioned …

具有全局中心化位移的离散蛇

Discrete snakes with globally centered displacements

本文证明了规模条件化临界Bienaymé树上全局中心化离散蛇的标度极限，在有限方差下收敛到Brownian蛇，在重尾位移下收敛到毛蛇变体，并应用于高度过程与Łukasiewicz路径及环树高度过程之差的标度极限。

Brownian disks and the Brownian snake

We provide a new construction of the Brownian disks, which have been defined by Bettinelli and Miermont as scaling limits of quadrangulations with a boundary when the boundary size tends to infinity. Our method is very similar to the construction of the Brownian map, but it makes use of the positive excursion measure of the Brownian snake which has been introduced recently. This excursion measure involves a random continuous tree whose vertices are assigned nonnegative labels, which correspond to distances from the boundary in our approach to the Brownian disk. We provide several applications …

临界分支随机游走及其蛇的标度极限

Scaling limits of critical branching random walks and their snakes

研究临界分支随机游走及其蛇的标度极限，在离散Sheu假设下证明重标度离散蛇收敛到ψ-布朗蛇，并应用于值域极限到反射ψ-布朗仙人掌。