The hull process of the Brownian plane

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• 作者：Nicolas Curien ; Jean-François Le Gall
• 关键词：Mathematics Subject Classification05C80 ; 60J68 ; 60J80
• 刊名：Probability Theory and Related Fields
• 出版年：2016
• 出版时间：October 2016
• 年：2016
• 卷：166
• 期：1-2
• 页码：187-231
• 全文大小：868 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Probability Theory and Stochastic Processes
  Mathematical and Computational Physics
  Quantitative Finance
  Mathematical Biology
  Statistics for Business, Economics, Mathematical Finance and Insurance
  Operation Research and Decision Theory
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-2064
• 卷排序：166

文摘

We study the random metric space called the Brownian plane, which is closely related to the Brownian map and is conjectured to be the universal scaling limit of many discrete random lattices such as the uniform infinite planar triangulation. We obtain a number of explicit distributions for the Brownian plane. In particular, we consider, for every \(r>0\), the hull of radius r, which is obtained by “filling in the holes” in the ball of radius r centered at the root. We introduce a quantity \(Z_r\) which is interpreted as the (generalized) length of the boundary of the hull of radius r. We identify the law of the process \((Z_r)_{r>0}\) as the time-reversal of a continuous-state branching process starting from \(+\infty \) at time \(-\infty \) and conditioned to hit 0 at time 0, and we give an explicit description of the process of hull volumes given the process \((Z_r)_{r>0}\). We obtain an explicit formula for the Laplace transform of the volume of the hull of radius r, and we also determine the conditional distribution of this volume given the length of the boundary. Our proofs involve certain new formulas for super-Brownian motion and the Brownian snake in dimension one, which are of independent interest.