The Problem of Predecessors on Spanning Trees

Authors

  • V. S. Poghosyan
  • V. B. Priezzhev

DOI:

https://doi.org/10.14311/1364

Keywords:

loop erased random walk, spanning trees, Kirchhoff theorem, Abelian sandpile model

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 to find 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) ∼ 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 states 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.

Downloads

Download data is not yet available.

Downloads

Published

2011-01-02

How to Cite

Poghosyan, V. S., & Priezzhev, V. B. (2011). The Problem of Predecessors on Spanning Trees. Acta Polytechnica, 51(2). https://doi.org/10.14311/1364

Issue

Section

Articles