随机平面格上的生成森林
Spanning Forests on Random Planar Lattices
摘要：
格点上生成森林的生成函数与q-态Potts模型在某个q -> 0极限下相关，并扩展了生成树或密集自避免支化聚合物的类似概念。最近的工作发现了一个组合扰动等价也与（二次作用）O（n）模型在极限n ->-1，扩展参数t计数的组件在森林中的数量。我们给出了一个随机矩阵制定的这个模型的集成度k随机平面晶格。当k = 3时，我们发现了与回路气体问题的Kostov解的对应关系，这是在n =-2时对（对数作用）O（n）模型的重新表述。然后，我们展示如何围绕t = 0理论进行展开。在热力学极限下，在t的任意阶，我们都有有限维柯西积分的有限和。主要的贡献来自一类特殊的条款，其中一个remation可以准确地执行。
英文摘要：
The generating function for spanning forests on a lattice is related to the q-state Potts model in a certain q -> 0 limit, and extends the analogous notion for spanning trees, or dense self-avoiding branched polymers. Recent works have found a combinatorial perturbative equivalence also with the (quadratic action) O(n) model in the limit n -> -1, the expansion parameter t counting the number of components in the forest. We give a random-matrix formulation of this model on the ensemble of degree-k random planar lattices. For k = 3, a correspondence is found with the Kostov solution of the loop-gas problem, which arise as a reformulation of the (logarithmic action) O(n) model, at n = -2. Then, we show how to perform an expansion around the t = 0 theory. In the thermodynamic limit, at any order in t we have a finite sum of finite-dimensional Cauchy integrals. The leading contribution comes from a peculiar class of terms, for which a resummation can be performed exactly.
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