随机映射上O（n）模型的嵌套循环递归算法
A recursive approach to the O(n) model on random maps via nested loops
摘要：
我们考虑四价映射的O（n）循环模型，并展示如何将其改写为没有循环的二分映射模型。这是一个组合分解的结果，该分解包括沿着它们的循环切割O（n）模型配置，使得每个基本块是一个可以具有任意偶数面度的映射。在诱导统计中，这些映射是根据玻尔兹曼分布绘制的，玻尔兹曼分布的参数（面权重）由固定点条件确定。特别地，我们证明了O（n）模型的稠密和稀疏临界点对应于具有大面的二分映射（即其度分布具有厚尾）。不动点条件的线性积分方程的重新表达，使我们能够探索模型的相图。特别是，我们确定这个相图正是最简单的版本的模型，其中的循环是“刚性”。几个概括的模型进行了讨论。
英文摘要：
We consider the O(n) loop model on tetravalent maps and show how to rephrase it into a model of bipartite maps without loops. This follows from a combinatorial decomposition that consists in cutting the O(n) model configurations along their loops so that each elementary piece is a map that may have arbitrary even face degrees. In the induced statistics, these maps are drawn according to a Boltzmann distribution whose parameters (the face weights) are determined by a fixed point condition. In particular, we show that the dense and dilute critical points of the O(n) model correspond to bipartite maps with large faces (i.e. whose degree distribution has a fat tail). The re-expression of the fixed point condition in terms of linear integral equations allows us to explore the phase diagram of the model. In particular, we determine this phase diagram exactly for the simplest version of the model where the loops are "rigid". Several generalizations of the model are discussed.
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