%0 Journal Article
%T Distance statistics in large toroidal maps
%A E. Guitter
%J Statistics
%D 2010
%I arXiv
%R 10.1088/1742-5468/2010/04/P04018
%X 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.
%U http://arxiv.org/abs/1003.0372v2