English

$SLE_6$ and 2-d critical bond percolation on the square lattice

Probability 2024-12-18 v6 Mathematical Physics Complex Variables math.MP

Abstract

Through the rotational invariance of the 2-d critical bond percolation exploration path on the square lattice we express Smirnov's edge parafermionic observable as a sum of two new edge observables. With the help of these two new edge observables we can apply the discrete harmonic analysis and conformal mapping theory to prove the convergence of the 2-d critical bond percolation exploration path on the square lattice to the trace of SLE6SLE_6 as the mesh size of the lattice tends to zero.

Keywords

Cite

@article{arxiv.2409.03235,
  title  = {$SLE_6$ and 2-d critical bond percolation on the square lattice},
  author = {Wang Zhou},
  journal= {arXiv preprint arXiv:2409.03235},
  year   = {2024}
}

Comments

Minor revision. 115 pages, 10 figures