Exact critical exponent for the shortest-path scaling function in percolation

TL;DR

This paper derives an exact value for the critical exponent related to shortest-path scaling in 2D percolation using crossing probability results, confirming previous simulation findings.

Contribution

It provides the first exact analytical determination of the critical exponent g_1 in 2D percolation, connecting crossing probabilities with shortest-path scaling.

Findings

Exact value g_1 = 25/24 in 2D percolation

Consistency with previous simulation results

Uses crossing probability from Cardy for derivation

Abstract

It is shown that the critical exponent $g_1$ related to pair-connectiveness and shortest-path (or chemical distance) scaling, recently studied by Porto et al., Dokholyan et al., and Grassberger, can be found exactly in 2d by using a crossing-probability result of Cardy, with the outcome $g_1 = 25/24$. This prediction is consistent with existing simulation results.