1203.4858 (Adrien Kassel et al.)

On the uniform cycle-rooted spanning tree in Z^2    [PDF]

Adrien Kassel, Richard Kenyon, Wei Wu

We compute the asymptotics of the first and second moments of the area of the cycle of a random cycle-rooted spanning tree (spanning unicycle) of any sequence of graphs $G_n\subset {\mathbb Z}^2$, such that $\frac{1}{n}G_n$ approximates a bounded domain $D\subset{\mathbb C}$. We show that the first and second moments grow like $\frac{4}{\pi}\log n$ and $C\cdot\text{Area}(D)n^2$, respectively, for an explicit constant $C=C(D)$. We use these results to give a lower bound for the first and third moments of the length of the random loop obtained by adding an independent random edge to a uniform spanning tree on $G_n$.

View original: http://arxiv.org/abs/1203.4858