标题： 从伸长的生成树到邪恶随机游走 显示英文标题

标题： From elongated spanning trees to vicious random walks

摘要： 给定一个大正方形格点上的生成森林，我们通过组合方法考虑沿树的分支的$k$条路径（$k$为奇数）的相关函数，或者等价地，$k$条环消除随机游走的相关函数。 路径的起点和终点以$k$腿水melon的方式分组。 对于起点和终点分组之间的距离$r$很大时，水melon配置数目与生成树总数的比值表现为$r^{-\nu} \log r$，其中$\nu = (k^2-1)/2$。 考虑沿着这个西瓜的子午线展开的生成森林，我们看到二维$k$-腿环消除西瓜指数$\nu$正在转化为 (1+1) 维有害行走者$k$在给定点的汇合概率的标度指数，$\tilde{\nu} = k^2/2$。 同时，我们表达了关于与可积系统可能关系的猜想。 显示英文摘要

摘要： Given a spanning forest on a large square lattice, we consider by combinatorial methods a correlation function of $k$ paths ($k$ is odd) along branches of trees or, equivalently, $k$ loop--erased random walks. Starting and ending points of the paths are grouped in a fashion a $k$--leg watermelon. For large distance $r$ between groups of starting and ending points, the ratio of the number of watermelon configurations to the total number of spanning trees behaves as $r^{-\nu} \log r$ with $\nu = (k^2-1)/2$. Considering the spanning forest stretched along the meridian of this watermelon, we see that the two--dimensional $k$--leg loop--erased watermelon exponent $\nu$ is converting into the scaling exponent for the reunion probability (at a given point) of $k$ (1+1)--dimensional vicious walkers, $\tilde{\nu} = k^2/2$. Also, we express the conjectures about the possible relation to integrable systems.