What is the probability of connecting two points ?

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 $0\le \rho \le 1$.

a few critical polynomials are at the end of the .tex source file

References in corpus (8)

• Statistical mechanics of complex networks
• The multivariate Tutte polynomial (alias Potts model) for graphs and matroids
• Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models I. General Theory and Square-Lattice Chromatic Polynomial
• Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models III. Triangular-Lattice Chromatic Polynomial
• The Brown-Colbourn conjecture on zeros of reliability polynomials is false
• Reliability Polynomials and their Asymptotic Limits for Families of Graphs
• Exact Partition Function for the Potts Model with Next-Nearest Neighbor Couplings on Strips of the Square Lattice
• Transfer Matrices for the Partition Function of the Potts Model on Toroidal Lattice Strips

Cited by in corpus (1)

• Fast Reliability Ranking of Matchstick Minimal Networks