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A view from infinity of the uniform infinite planar quadrangulation. (English) Zbl 1277.05151
Summary: We introduce a new construction of the uniform infinite planar quadrangulation (UIPQ). Our approach is based on an extension of the Cori-Vauquelin-Schaeffer mapping in the context of infinite trees, in the spirit of previous work P. Chassaing and B. Durhuus [Ann. Probab. 34, No. 3, 879–917 (2006; Zbl 1102.60007)], J.-F. Le Gall and L. Ménard [Ill. J. Math. 54, No. 3, 1163–1203 (2010; Zbl 1259.60035)] and L. Ménard [Ann. Inst. Henri Poincaré, Probab. Stat. 46, No. 1, 190–208 (2010; Zbl 1201.60009)]. However, we release the positivity constraint on the labels of trees which was imposed in these references, so that our construction is technically much simpler. This approach allows us to prove the conjectures of M. Krikun [“On one property of distances in the infinite random quadrangulation”, arXiv: 0805.1907] pertaining to the “geometry at infinity” of the UIPQ, and to derive new results about the UIPQ, among which a fine study of infinite geodesics.

MSC:
05C80 Random graphs (graph-theoretic aspects)
05C63 Infinite graphs
05C05 Trees
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60J68 Superprocesses
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