The problem of predecessors on spanning trees

TL;DR

This paper studies the probability of paths passing through fixed sites on spanning trees on a square lattice, using conformal field theory, analytical methods, and simulations to reveal decay rates and specific probabilities.

Contribution

It provides a theoretical prediction for the decay of predecessor probabilities and exact probabilities for neighboring sites, supported by Monte Carlo simulations.

Findings

Probability decreases as r^{-3/4} for large distances

Probability for neighboring sites is 5/16

Simulations confirm theoretical predictions

Abstract

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.