From elongated spanning trees to vicious random walks

A. Gorsky 1, S. Nechaev 23, V. S. Poghosyan 4, V. B. Priezzhev 5

Nuclear Physics B 870 (2013) 55-77

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 in a fashion 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 see 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, $\tilde{\nu} = k^2/2$. Also, we express the conjectures about the possible relation to integrable systems.

  • 1 : ITEP
    ITEP
  • 2 : Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS)
    CNRS : UMR8626 – Université Paris XI - Paris Sud
  • 3 : P.N. Lebedev Physical Institute of the Russian Academy of Sciences
    Russian Academy of Science
  • 4 : Institute for Informatics and Automation Problems
    NAS of Armenia,
  • 5 : Bogolubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research
    Russian Academy of Science