### 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.

Original language | English (US) |
---|---|

Title of host publication | Stochastic Geometry, Spatial Statistics and Random Fields |

Subtitle of host publication | Asymptotic Methods |

Publisher | Springer-Verlag |

Pages | 399-420 |

Number of pages | 22 |

ISBN (Print) | 9783642333040 |

DOIs | |

State | Published - Jan 1 2013 |

### Publication series

Name | Lecture Notes in Mathematics |
---|---|

Volume | 2068 |

ISSN (Print) | 0075-8434 |

ISSN (Electronic) | 1617-9692 |

### Fingerprint

### Keywords

- Brownian Sheet
- Gibbs Distribution
- Large Deviation Principle
- Random Tree
- Stochastic Partial Differential Equation

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Stochastic Geometry, Spatial Statistics and Random Fields: Asymptotic Methods*(pp. 399-420). (Lecture Notes in Mathematics; Vol. 2068). Springer-Verlag. https://doi.org/10.1007/978-3-642-33305-7_12

**Geometry of Large Random Trees : SPDE Approximation.** / Bakhtin, Yuri.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

*Stochastic Geometry, Spatial Statistics and Random Fields: Asymptotic Methods.*Lecture Notes in Mathematics, vol. 2068, Springer-Verlag, pp. 399-420. https://doi.org/10.1007/978-3-642-33305-7_12

}

TY - CHAP

T1 - Geometry of Large Random Trees

T2 - SPDE Approximation

AU - Bakhtin, Yuri

PY - 2013/1/1

Y1 - 2013/1/1

N2 - 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.

AB - 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.

KW - Brownian Sheet

KW - Gibbs Distribution

KW - Large Deviation Principle

KW - Random Tree

KW - Stochastic Partial Differential Equation

UR - http://www.scopus.com/inward/record.url?scp=85072862004&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85072862004&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-33305-7_12

DO - 10.1007/978-3-642-33305-7_12

M3 - Chapter

AN - SCOPUS:85072862004

SN - 9783642333040

T3 - Lecture Notes in Mathematics

SP - 399

EP - 420

BT - Stochastic Geometry, Spatial Statistics and Random Fields

PB - Springer-Verlag

ER -