What is the probability of connecting two points ?

TL;DR

This paper explores the mathematical modeling of two-terminal reliability in percolation theory, expressing it through transfer matrices and analyzing its asymptotic behavior and structural transitions based on link and site probabilities.

Contribution

It introduces a transfer matrix approach to exactly compute two-terminal reliability considering link and site probabilities, revealing asymptotic power-law behavior and structural transitions.

Findings

Reliability obeys a power-law asymptotic behavior

Largest eigenvalue of transfer matrix determines scaling

Complex zeros exhibit structural transitions

Abstract

The two-terminal reliability, known as the pair connectedness or connectivity function in percolation theory, may actually be expressed as a product of transfer matrices in which the probability of operation of each link and site is exactly taken into account. When link and site probabilities are $p$ and $\rho$, it obeys an asymptotic power-law behavior, for which the scaling factor is the transfer matrix's eigenvalue of largest modulus. The location of the complex zeros of the two-terminal reliability polynomial exhibits structural transitions as $0 \leq \rho \leq 1$.