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Local Convergence of Large Critical Multi-type Galton–Watson Trees and Applications to Random Maps

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  • Robin Stephenson

    (Universität Zürich)

Abstract

We show that large critical multi-type Galton–Watson trees, when conditioned to be large, converge locally in distribution to an infinite tree which is analogous to Kesten’s infinite monotype Galton–Watson tree. This is proven when we condition on the number of vertices of one fixed type, and with an extra technical assumption if we count at least two types. We then apply these results to study local limits of random planar maps, showing that large critical Boltzmann-distributed random maps converge in distribution to an infinite map.

Suggested Citation

  • Robin Stephenson, 2018. "Local Convergence of Large Critical Multi-type Galton–Watson Trees and Applications to Random Maps," Journal of Theoretical Probability, Springer, vol. 31(1), pages 159-205, March.
    • Handle: RePEc:spr:jotpro:v:31:y:2018:i:1:d:10.1007_s10959-016-0707-3
    DOI: 10.1007/s10959-016-0707-3
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    Cited by:

    1. Xin He, 2022. "Local Convergence of Critical Random Trees and Continuous-State Branching Processes," Journal of Theoretical Probability, Springer, vol. 35(2), pages 685-713, June.
    2. Benedikt Stufler, 2022. "Quenched Local Convergence of Boltzmann Planar Maps," Journal of Theoretical Probability, Springer, vol. 35(2), pages 1324-1342, June.
    3. Benedikt Stufler, 2022. "Rerooting Multi-type Branching Trees: The Infinite Spine Case," Journal of Theoretical Probability, Springer, vol. 35(2), pages 653-684, June.

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