Séminaire du LPTMS: Serguei Nechaev


11:00 - 12:00

LPTMS, salle 201, 2ème étage, Bât 100, Campus d'Orsay
15 Rue Georges Clemenceau, Orsay, 91405
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From elongated spanning trees to vicious random walks

Serguei Nechaev, LPTMS

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^{-\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. Some consequences and generalizations of this result will be also discussed.

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