Universality for bond percolation in two dimensions

TL;DR

This paper proves that various bond percolation models on common 2D lattices share the same critical exponents, establishing a universality class using the star-triangle transformation and box-crossing property.

Contribution

It demonstrates universality of critical exponents for all (in)homogeneous bond percolation models on key lattices in two dimensions, extending previous results.

Findings

Models have identical critical exponents at the critical point.

Universality applies to multiple exponents including one-arm, volume, and connectivity.

Results rely on star-triangle transformation and box-crossing property.

Abstract

All (in)homogeneous bond percolation models on the square, triangular, and hexagonal lattices belong to the same universality class, in the sense that they have identical critical exponents at the critical point (assuming the exponents exist). This is proved using the star-triangle transformation and the box-crossing property. The exponents in question are the one-arm exponent $\rho$, the 2j-alternating-arms exponents $\rho_{2j}$ for $j\ge1$, the volume exponent $\delta$, and the connectivity exponent $\eta$. By earlier results of Kesten, this implies universality also for the near-critical exponents $\beta$, $\gamma$, $\nu$, $\Delta$ (assuming these exist) for any of these models that satisfy a certain additional hypothesis, such as the homogeneous bond percolation models on these three lattices.