Critical percolation and lack of self-averaging in disordered models

TL;DR

This paper argues that critical two-dimensional percolation on the square lattice does not exhibit self-averaging, due to the persistent sample dependence of phase space fragmentation in disordered models.

Contribution

It provides new theoretical insights from percolation theory supporting the conjecture of non-self-averaging at criticality in 2D percolation.

Findings

Critical 2D percolation lacks self-averaging.

Sample dependence persists in the thermodynamic limit.

Fragmentation of phase space explains non-self-averaging.

Abstract

Lack of self-averaging originates in many disordered models from a fragmentation of the phase space where the sizes of the fragments remain sample-dependent in the thermodynamic limit. On the basis of new results in percolation theory, we give here an argument in favour of the conjecture that critical two dimensional percolation on the square lattice lacks of self-averaging.