From elongated spanning trees to vicious random walks

Abstract

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.

27 pages, 6 figures

References in corpus (6)

• Non-intersecting Brownian walkers and Yang-Mills theory on the sphere
• Indecomposability parameters in chiral Logarithmic Conformal Field Theory
• Logarithmic observables in critical percolation
• O'Connell's process as a vicious Brownian motion
• Survival probability of mutually killing Brownian motions and the O'Connell process
• Random ballistic growth and diffusion in symmetric spaces

Cited by in corpus (3)

• Correlations in the $n\to 0$ limit of the dense O(n) loop model
• Watermelons on the half-plane
• Dissipative Abelian Sandpile Models