On surface properties of two-dimensional percolation clusters

TL;DR

This study uses transfer-matrix methods to analyze surface properties of 2D percolation clusters, accurately determining critical exponents and confirming theoretical predictions about surface decay and transition behavior.

Contribution

It provides precise numerical estimates of surface critical exponents in 2D percolation, confirming conformal invariance predictions and clarifying the nature of surface transitions.

Findings

Surface decay-of-correlations exponent η_s ≈ 2/3

No special transition occurs in the studied case

Irrelevant exponent y_s ≈ -1 at the ordinary transition

Abstract

The two-dimensional site percolation problem is studied by transfer-matrix methods on finite-width strips with free boundary conditions. The relationship between correlation-length amplitudes and critical indices, predicted by conformal invariance, allows a very precise determination of the surface decay-of-correlations exponent, $\eta_s = 0.6664 \pm 0.0008$, consistent with the analytical value $\eta_s = 2/3$. It is found that a special transition does not occur in the case, corroborating earlier series results. At the ordinary transition, numerical estimates are consistent with the exact value $y_s = -1$ for the irrelevant exponent.