Geodesic distance in planar graphs
Geodesic distance in planar graphs
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DOI：
10.1016/s0550-3213(03)00355-9
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发表时间：
2003-07-28
影响因子：
2.8
通讯作者：
Guitter, E
中科院分区：
文献类型：
--
作者：
Bouttier, J;Di Francesco, P;Guitter, E
我们导出了具有任意偶价顶点和在固定测地线距离上有两个标记点的平面地图（零属图）的精确生成函数。这是以一种纯粹的组合方式完成的，基于带有装饰树的双射，导致测地线距离的递归关系。后者可以用离散的类孤子表达式精确地求解，暗示了一个潜在的可积结构。我们从该解中提取了各（多）临界点处的分形维数，以及连续两点函数的精确标度形式和（多）临界随机曲面测地线距离的概率分布。两点函数服从涉及KdV层次残数的微分方程。(C) 2003 Elsevier B.V.版权所有
We derive the exact generating function for planar maps (genus zero fatgraphs) with vertices of arbitrary even valence and with two marked points at a fixed geodesic distance. This is done in a purely combinatorial way based on a bijection with decorated trees, leading to a recursion relation on the geodesic distance. The latter is solved exactly in terms of discrete soliton-like expressions, suggesting an underlying integrable structure. We extract from this solution the fractal dimensions at the various (multi)-critical points, as well as the precise scaling forms of the continuum two-point functions and the probability distributions for the geodesic distance in (multi)-critical random surfaces. The two-point functions are shown to obey differential equations involving the residues of the KdV hierarchy. (C) 2003 Elsevier B.V. All rights reserved.