2010/03/31 by E. Guitter, E Guitter
Computer Science · Mathematics · Physics and Astronomy · #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #Topological and Geometric Data Analysis #math-ph #math.CO #math.MP #math.PR
paper · pdf · doi:10.1088/1742-5468/2010/04/p04018
published as J. Stat. Mech. 2010 P04018 (2010) · 24 pages, 9 figures, minor corrections, new added references
openalex publication_date 2010/04/21 · arxiv created 2010/06/30 · arxiv updated 2010/07/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/30
We compute a number of distance-dependent universal scaling functions characterizing the distance statistics of large maps of genus one. In particular, we obtain explicitly the probability distribution for the length of the shortest non-contractible loop passing via a random point in the map, and that for the distance between two random points. Our results are derived in the context of bipartite toroidal quadrangulations, using their coding by well-labeled 1-trees, which are maps of genus one with a single face and appropriate integer vertex labels. Within this framework, the distributions above are simply obtained as scaling limits of appropriate generating functions for well-labeled 1-trees, all expressible in terms of a small number of basic scaling functions for well-labeled plane trees.