From elongated spanning trees to vicious random walks

TL;DR

This paper explores the combinatorial properties of spanning forests and loop-erased random walks, revealing new scaling exponents and conjecturing links to integrable systems, with implications for understanding vicious walkers.

Contribution

It introduces a novel combinatorial approach to analyze correlation functions of paths in spanning forests and connects these to vicious random walks and integrable systems.

Findings

Derived the asymptotic behavior of watermelon configurations as r^{- u} log r.

Identified the relation between loop-erased walk exponents and vicious walker reunion probabilities.

Proposed conjectures linking these models to integrable systems.

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.