# 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.
Document type :
Journal articles
Domain :

https://hal.archives-ouvertes.fr/hal-00805989
Contributor : Claudine Le Vaou <>
Submitted on : Friday, March 29, 2013 - 12:14:24 PM
Last modification on : Wednesday, October 14, 2020 - 4:00:51 AM

### Identifiers

• HAL Id : hal-00805989, version 1
• ARXIV : 1206.3147

### Citation

A. Gorsky, S. Nechaev, V. S. Poghosyan, V. B. Priezzhev. From elongated spanning trees to vicious random walks. Nuclear Physics B, Elsevier, 2013, 870, pp.55-77. ⟨hal-00805989⟩

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