Marckert, Jean-François; Miermont, Grégory Invariance principles for random bipartite planar maps. (English) Zbl 1208.05135 Ann. Probab. 35, No. 5, 1642-1705 (2007). Summary: Random planar maps are considered in the physics literature as the discrete counterpart of random surfaces. It is conjectured that properly rescaled random planar maps, when conditioned to have a large number of faces, should converge to a limiting surface whose law does not depend, up to scaling factors, on details of the class of maps that are sampled. Previous works on the topic, starting with Chassaing and Schaeffer, have shown that the radius of a random quadrangulation with \(n\) faces, that is, the maximal graph distance on such a quadrangulation to a fixed reference point, converges in distribution once rescaled by \(n^{1/4}\) to the diameter of the Brownian snake, up to a scaling constant.Using a bijection due to Bouttier, Di Francesco and Guitter between bipartite planar maps and a family of labeled trees, we show the corresponding invariance principle for a class of random maps that follow a Boltzmann distribution putting weight \(q_k\) on faces of degree \(2k\): the radius of such maps, conditioned to have \(n\) faces (or \(n\) vertices) and under a criticality assumption, converges in distribution once rescaled by \(n^{1/4}\) to a scaled version of the diameter of the Brownian snake. Convergence results for the so-called profile of maps are also provided. The convergence of rescaled bipartite maps to the Brownian map, in the sense introduced by Marckert and Mokkadem, is also shown. The proofs of these results rely on a new invariance principle for two-type spatial Galton-Watson trees. Cited in 1 ReviewCited in 37 Documents MSC: 05C80 Random graphs (graph-theoretic aspects) 05C30 Enumeration in graph theory 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 60C05 Combinatorial probability 60F17 Functional limit theorems; invariance principles Keywords:Random planar maps; labeled mobiles; invariance principle; spatial Galton–Watson trees; Brownian snake; Brownian map × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Aldous, D. J. (1991). The continuum random tree. II. An overview. In Stochastic Analysis ( Durham , 1990 ) . London Math. Soc. Lecture Note Ser. 167 23–70. Cambridge Univ. Press. · Zbl 0791.60008 [2] Aldous, D. J. (1993). The continuum random tree. III. Ann. 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