Recurrence of bipartite planar maps. (English) Zbl 1286.05153

Björnberg, Jakob Erik; Stefánsson, Sigurdur Örn

Electron. J. Probab. 19, Paper No. 31, 40 p. (2014).

Summary: This paper concerns random bipartite planar maps which are defined by assigning weights to their faces. The paper presents a threefold contribution to the theory. Firstly, we prove the existence of the local limit for all choices of weights and describe it in terms of an infinite mobile. Secondly, we show that the local limit is in all cases almost surely recurrent. And thirdly, we show that for certain choices of weights the local limit has exactly one face of infinite degree and has in that case spectral dimension 4/3 (the latter requires a mild moment condition).

MSC:

05C80	Random graphs (graph-theoretic aspects)
05C81	Random walks on graphs
05C05	Trees
60J80	Branching processes (Galton-Watson, birth-and-death, etc.)
60F05	Central limit and other weak theorems

Keywords:

planar maps; local limits; simply generated trees; random walk