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Short cycles in high genus unicellular maps. (English) Zbl 1493.05085

Summary: We study large uniform random maps with one face whose genus grows linearly with the number of edges, which are a model of discrete hyperbolic geometry. In previous works, several hyperbolic geometric features have been investigated. In the present work, we study the number of short cycles in a uniform unicellular map of high genus, and we show that it converges to a Poisson distribution. As a corollary, we obtain the law of the systole of uniform unicellular maps in high genus. We also obtain the asymptotic distribution of the vertex degrees in such a map.

MSC:

05C12 Distance in graphs
05C38 Paths and cycles
05C10 Planar graphs; geometric and topological aspects of graph theory
05C80 Random graphs (graph-theoretic aspects)
60C05 Combinatorial probability
60D05 Geometric probability and stochastic geometry
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References:

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