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Walks and Paths in Trees
Recently Csikv\'ari \cite{csik} proved a conjecture of Nikiforov concerning
the number of closed walks on trees. Our aim is to extend his theorem to all
walks. In addition, we give a simpler proof of Csikv\'ari's result and answer
one of his questions in the negative. Finally we consider an analogous question
for paths rather than walks
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 paths ( is odd) along
branches of trees or, equivalently, loop--erased random walks. Starting and
ending points of the paths are grouped in a fashion a --leg watermelon. For
large distance between groups of starting and ending points, the ratio of
the number of watermelon configurations to the total number of spanning trees
behaves as with . Considering the spanning
forest stretched along the meridian of this watermelon, we see that the
two--dimensional --leg loop--erased watermelon exponent is converting
into the scaling exponent for the reunion probability (at a given point) of
(1+1)--dimensional vicious walkers, . Also, we express the
conjectures about the possible relation to integrable systems.Comment: 27 pages, 6 figure
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