两种四边形统计集成的比较：一种连分数方法
Comparing two statistical ensembles of quadrangulations: a continued fraction approach
摘要：
我们使用一个连分数的方法来比较两个统计集成的四边形与边界，都由两个参数控制。在第一个集成中，四边形是双色的，并且参数控制它们的两种颜色的顶点的数量。在第二个集合中，参数控制的是对于到给定顶点的距离是局部最大值的顶点的数量，以及不是局部最大值的顶点的数量。这两个系综都可以通过它们在固定边界长度处的（二元）生成函数来描述，或者在一些标准切片分解之后，通过它们的（二元）切片生成函数来描述。我们首先表明，固定边界长度的生成函数实际上是相等的两个合奏。然后，我们表明，切片生成函数，虽然不同的两个合奏，简单地对应于两种不同的方式编码相同的数量作为一个连分数。这个属性是用来获得显式表达式的切片生成函数的建设性的方式。
英文摘要：
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.
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