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

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.

References updated, small corrections 4 pages. Published version

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• Efficient simulation of the random-cluster model
• Dependence of Conductance on Percolation Backbone Mass
• Exact results for some Madelung type constants in the finite-size scaling theory
• Fractal Behavior of the Shortest Path Between Two Lines in Percolation Systems
• A Simple Five-Dimensional Wave Equation for a Dirac Particle
• On the universality of distribution of ranked cluster masses at critical percolation