Comparing two statistical ensembles of quadrangulations: a continued fraction approach

Fusy, Éric; Guitter, Emmanuel

Annales de l’Institut Henri Poincaré D, Tome 4 (2017) no. 2, pp. 125-176.

We use a continued fraction approach to compare two statistical ensembles of quadrangulations with a boundary, both controlled by two parameters. In the fi�rst ensemble, the quadrangulations are bicolored and the parameters control their numbers of vertices of both colors. In the second ensemble, the parameters control instead the number of vertices which are local maxima for the distance to a given vertex, and the number of those which are not. Both ensembles may be described either by their (bivariate) generating functions at �fixed boundary length or, after some standard slice decomposition, by their (bivariate) slice generating functions. We fi�rst show that the �fixed boundary length generating functions are in fact equal for the two ensembles. We then show that the slice generating functions, although di�fferent for the two ensembles, simply correspond to two di�fferent ways of encoding the same quantity as a continued fraction. This property is used to obtain explicit expressions for the slice generating functions in a constructive way.

Classification : 05-XX, 00-XX
Mots-clés : Planar maps, quadrangulations, two-point functions, continued fractions, hard dimers, heaps theory