From elongated spanning trees to vicious random walks
Sergei Nechaev
LPTMS, Orsay
Mon, Dec. 09th 2013, 11:00
Salle Claude Itzykson, Bât. 774, Orme des Merisiers
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 this watermelon, we show 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, $tilde{nu} = k^2/2$.
Contact : Vincent PASQUIER


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