Equality of bond percolation critical exponents for pairs of dual lattices

TL;DR

This paper proves that for certain two-dimensional lattice pairs, the bond percolation critical exponents are identical, using a computational and generalized substitution method, expanding understanding of universality in percolation theory.

Contribution

It introduces a new proof technique for the equality of critical exponents in lattice-dual pairs, applicable to a broad class of two-dimensional lattices.

Findings

Critical exponents are equal for lattice-dual pairs in the studied class.

The substitution method effectively proves exponent equality without extensive computation.

An infinite collection of lattice sets share the same critical exponents.

Abstract

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.