From elongated spanning trees to vicious random walks

Given a spanning forest on a large square lattice, we consider by Kirchhoff theorem 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 $r^{-\nu}\log r$ with $\nu = (k^2-1)/2$. Considering the spanning forest stretched along the meridian of the watermelon, we show that the two-dimensional $k$-leg loop-erased watermelon exponent $\nu$, corresponding to $c=-2$ CFT is converting into the scaling exponent for the reunion probability (at a given point) of $k$ $(1+1)$-dimensional vicious walkers, $\tilde{\nu} = k^2/2$, described by RMT.