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November 2019 Scaling limit of random forests with prescribed degree sequences
Tao Lei
Bernoulli 25(4A): 2409-2438 (November 2019). DOI: 10.3150/18-BEJ1058

Abstract

In this paper, we consider the random plane forest uniformly drawn from all possible plane forests with a given degree sequence. Under suitable conditions on the degree sequences, we consider the limit of a sequence of such forests with the number of vertices tends to infinity in terms of Gromov–Hausdorff–Prokhorov topology. This work falls into the general framework of showing convergence of random combinatorial structures to certain Gromov–Hausdorff scaling limits, described in terms of the Brownian Continuum Random Tree (BCRT), pioneered by the work of Aldous (Ann. Probab. 19 (1991) 1–28; In Stochastic Analysis (Durham, 1990) (1991) 23–70 Cambridge Univ. Press; Ann. Probab. 21 (1993) 248–289). In fact, we identify the limiting random object as a sequence of random real trees encoded by excursions of some first passage bridges reflected at minimum. We establish such convergence by studying the associated Lukasiewicz walk of the degree sequences. In particular, our work is closely related to and uses the results from the recent work of Broutin and Marckert (Random Structures Algorithms 44 (2014) 290–316) on scaling limit of random trees with prescribed degree sequences, and the work of Addario-Berry (Random Structures Algorithms 41 (2012) 253–261) on tail bounds of the height of a random tree with prescribed degree sequence.

Citation

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Tao Lei. "Scaling limit of random forests with prescribed degree sequences." Bernoulli 25 (4A) 2409 - 2438, November 2019. https://doi.org/10.3150/18-BEJ1058

Information

Received: 1 October 2017; Revised: 1 March 2018; Published: November 2019
First available in Project Euclid: 13 September 2019

zbMATH: 07110100
MathSciNet: MR4003553
Digital Object Identifier: 10.3150/18-BEJ1058

Rights: Copyright © 2019 Bernoulli Society for Mathematical Statistics and Probability

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Vol.25 • No. 4A • November 2019
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