Solution of the One-Dimensional Bond Problem in a Percolation Theory

TL;DR

This paper investigates key properties of one-dimensional percolation theory, revealing that finite systems satisfy stability conditions but scaling hypotheses do not hold, providing new insights into bond and site percolation behaviors.

Contribution

It demonstrates for the first time that the scaling hypothesis is invalid for one-dimensional bond percolation in finite systems, while stability conditions are met.

Findings

Finite-size systems satisfy stability conditions

Scaling hypothesis is invalid for 1D bond percolation

Critical exponents are characterized for bonds and sites

Abstract

The results of investigations of main characteristics of a one-dimensional percolation theory (percolation threshold, critical exponents of correlation radius and specific heat, and free energy) are presented for the problem of bonds and sites. For the first time it is shown that for a finite-size system the stability condition is fulfilled while the scaling hypothesis is inacceptable for one-dimensional bond problem.