### Abstract

We consider Gibbs distributions on finite random plane trees with bounded branching. We show that as the order of the tree grows to infinity, the distribution of any finite neighborhood of the root of the tree converges to a limit. We compute the limiting distribution explicitly and study its properties. We introduce an infinite random tree consistent with these limiting distributions and show that it satisfies a certain form of the Markov property. We also study the growth of this tree and prove several limit theorems including a diffusion approximation.

Original language | English (US) |
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Pages (from-to) | 312-331 |

Number of pages | 20 |

Journal | Random Structures and Algorithms |

Volume | 37 |

Issue number | 3 |

DOIs | |

State | Published - Oct 2010 |

### Fingerprint

### Keywords

- Gibbs distribution
- Infinite volume limit
- Random trees

### ASJC Scopus subject areas

- Computer Graphics and Computer-Aided Design
- Software
- Mathematics(all)
- Applied Mathematics

### Cite this

**Thermodynamic limit for large random trees.** / Bakhtin, Yuri.

Research output: Contribution to journal › Article

*Random Structures and Algorithms*, vol. 37, no. 3, pp. 312-331. https://doi.org/10.1002/rsa.20317

}

TY - JOUR

T1 - Thermodynamic limit for large random trees

AU - Bakhtin, Yuri

PY - 2010/10

Y1 - 2010/10

N2 - We consider Gibbs distributions on finite random plane trees with bounded branching. We show that as the order of the tree grows to infinity, the distribution of any finite neighborhood of the root of the tree converges to a limit. We compute the limiting distribution explicitly and study its properties. We introduce an infinite random tree consistent with these limiting distributions and show that it satisfies a certain form of the Markov property. We also study the growth of this tree and prove several limit theorems including a diffusion approximation.

AB - We consider Gibbs distributions on finite random plane trees with bounded branching. We show that as the order of the tree grows to infinity, the distribution of any finite neighborhood of the root of the tree converges to a limit. We compute the limiting distribution explicitly and study its properties. We introduce an infinite random tree consistent with these limiting distributions and show that it satisfies a certain form of the Markov property. We also study the growth of this tree and prove several limit theorems including a diffusion approximation.

KW - Gibbs distribution

KW - Infinite volume limit

KW - Random trees

UR - http://www.scopus.com/inward/record.url?scp=77957325027&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77957325027&partnerID=8YFLogxK

U2 - 10.1002/rsa.20317

DO - 10.1002/rsa.20317

M3 - Article

VL - 37

SP - 312

EP - 331

JO - Random Structures and Algorithms

JF - Random Structures and Algorithms

SN - 1042-9832

IS - 3

ER -