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Geometry of Large Random Trees: SPDE Approximation

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Stochastic Geometry, Spatial Statistics and Random Fields

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2068))

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Abstract

In this chapter we present a point of view at large random trees. We study the geometry of large random rooted plane trees under Gibbs distributions with nearest neighbour interaction. In the first section of this chapter, we study the limiting behaviour of the trees as their size grows to infinity. We give results showing that the branching type statistics is deterministic in the limit, and the deviations from this law of large numbers follow a large deviation principle. Under the same limit, the distribution on finite trees converges to a distribution on infinite ones. These trees can be interpreted as realizations of a critical branching process conditioned on non-extinction. In the second section, we consider a natural embedding of the infinite tree into the two-dimensional Euclidean plane and obtain a scaling limit for this embedding. The geometry of the limiting object is of particular interest. It can be viewed as a stochastic foliation, a flow of monotone maps, or as a solution to a certain Stochastic PDE with respect to a Brownian sheet. We describe these points of view and discuss a natural connection with superprocesses.

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Authors and Affiliations

  1. Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Yuri Bakhtin

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  1. Yuri Bakhtin
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Correspondence to Yuri Bakhtin .

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  1. Helmholtzstr. 18, Ulm, 89069, Germany

    Evgeny Spodarev

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Bakhtin, Y. (2013). Geometry of Large Random Trees: SPDE Approximation. In: Spodarev, E. (eds) Stochastic Geometry, Spatial Statistics and Random Fields. Lecture Notes in Mathematics, vol 2068. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33305-7_12

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