The problem of predecessors on spanning trees

We consider the equiprobable distribution of spanning trees on the square lattice. All bonds of each tree can be oriented uniquely with respect to an arbitrary chosen site called the root. The problem of predecessors is finding the probability that a path along the oriented bonds passes sequentially fixed sites $i$ and $j$. The conformal field theory for the Potts model predicts the fractal dimension of the path to be 5/4. Using this result, we show that the probability in the predecessors problem for two sites separated by large distance $r$ decreases as $P(r) \sim r^{-3/4}$. If sites $i$ and $j$ are nearest neighbors on the square lattice, the probability $P(1)=5/16$ can be found from the analytical theory developed for the sandpile model. The known equivalence between the loop erased random walk (LERW) and the directed path on the spanning tree says that $P(1)$ is the probability for the LERW started at $i$ to reach the neighboring site $j$. By analogy with the self-avoiding walk, $P(1)$ can be called the return probability. Extensive Monte-Carlo simulations confirm the theoretical predictions.Comment: 7 pages, 2 figure

Exact Height Probabilities in the Abelian Sandpile Model

We study Bak, Tang and Wiesenfeld’s Abelian sandpile model of self-organized criticality on 2D square lattice. A combinatorial method for evaluation of height probabilities is proposed. Exact analytical expression for the fractional number of sites having height 2 is obtained

Critical Dynamics of Self-Organizing Eulerian Walkers

The model of self-organizing Eulerian walkers is numerically investigated on the square lattice. The critical exponents for the distribution of a number of steps ($\tau_l$) and visited sites ($\tau_s$) characterizing the process of transformation from one recurrent configuration to another are calculated using the finite-size scaling analysis. Two different kinds of dynamical rules are considered. The results of simulations show that both the versions of the model belong to the same class of universality with the critical exponents $\tau_l=\tau_s=1.75\pm 0.1$.Comment: 3 pages, 4 Postscript figures, RevTeX, additional information available at http://thsun1.jinr.dubna.su/~shche