2009/03/21 by Matthew R. A. Sedlock, John C. Wierman
Mathematics · Physics and Astronomy · #Algorithm #Class (philosophy) #Combinatorics #Computation #Computer science #Continuum percolation theory #Critical exponent #Directed percolation #Discrete mathematics #Geometry #Lattice (music) #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Mathematics #Percolation (cognitive psychology) #Percolation critical exponents #Percolation threshold #Physics #Quantum mechanics #Scaling #Statistical physics #Stochastic processes and statistical mechanics #cond-mat.stat-mech
paper · pdf · doi:10.1103/physreve.79.051119
10 pages, 7 figures
arxiv created 2009/03/21 · openalex publication_date 2009/05/19 · arxiv updated 2015/05/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a certain class of two-dimensional lattices, lattice-dual pairs are shown to have the same bond-percolation critical exponents. A computational proof is given for the martini lattice and its dual to illustrate the method. The result is generalized to a class of lattices that allows the equality of bond-percolation critical exponents for lattice-dual pairs to be concluded without performing the computations. The proof uses the substitution method, which involves stochastic ordering of probability measures on partially ordered sets. As a consequence, there is an infinite collection of infinite sets of two-dimensional lattices, such that all lattices in a set have the same critical exponents.