From elongated spanning trees to vicious random walks

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 such that they form 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−νlogr with ν=(k2−1)/2ν=(k2−1)/2. Considering the spanning forest stretched along the meridian of this watermelon, we show that the two-dimensional k-leg loop-erased watermelon exponent ν is converting into the scaling exponent for the reunion probability (at a given point) of k  (1+1)(1+1)-dimensional vicious walkers, ν˜=k2/2. At the end, we express the conjectures about the possible relation to integrable systems.