Comparing two statistical ensembles of quadrangulations: a continued fraction approach

TL;DR

This paper compares two statistical ensembles of quadrangulations with boundaries using continued fractions, revealing their generating functions are related through different encodings of the same underlying quantity.

Contribution

It introduces a continued fraction approach to relate and explicitly compute the slice generating functions of two different quadrangulation ensembles.

Findings

Fixed boundary length generating functions are equal for both ensembles.

Slice generating functions differ but encode the same quantity via different continued fractions.

Explicit expressions for slice generating functions are obtained constructively.

Abstract

We use a continued fraction approach to compare two statistical ensembles of quadrangulations with a boundary, both controlled by two parameters. In the first 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 first 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 different for the two ensembles, simply correspond to two different 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.