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Local convergence of large critical multi-type Galton-Watson trees and applications to random maps. (English) Zbl 1393.05244
Summary: 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.

05C80 Random graphs (graph-theoretic aspects)
05C05 Trees
05C63 Infinite graphs
05C81 Random walks on graphs
60B10 Convergence of probability measures
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