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Percolation critical polynomial as a graph invariant.

Christian R Scullard¹

¹Lawrence Livermore National Laboratory, Livermore, California 94550, USA. scullard1@llnl.gov

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|December 11, 2012

Summary

This study introduces a generalized critical polynomial for bond percolation on lattices. This polynomial, computable via a recursive algorithm, offers accurate approximations for unknown critical probabilities, demonstrated on the kagome lattice.

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Area of Science:

Statistical Mechanics
Graph Theory
Computational Physics

Background:

The bond percolation critical probability, a key parameter in statistical mechanics, can be exactly determined for certain lattices via a critical polynomial.
A generalized critical polynomial has been developed for any periodic lattice, dependent on lattice structure and subgraph decomposition.
This generalized polynomial approximates unknown critical probabilities, with accuracy improving as subgraph size increases.

Purpose of the Study:

To demonstrate that the generalized critical polynomial can be treated as a graph invariant, analogous to the Tutte polynomial.
To establish a computational method for calculating the generalized critical polynomial on finite graphs.
To apply this method to estimate the percolation threshold for the kagome lattice.

Main Methods:

The generalized critical polynomial is computed on finite graphs using the recursive deletion-contraction algorithm.
This algorithm allows for computer-based calculations of the polynomial.
The method is applied to the kagome lattice using subgraphs up to 36 bonds.

Main Results:

The generalized critical polynomial is shown to be a graph invariant, computable via deletion-contraction.
Calculations for the kagome lattice yield a predicted critical probability of p(c)=0.52440572....
This prediction closely matches the known numerical value, with a difference of only 6.9×10(-7).

Conclusions:

The generalized critical polynomial serves as a valuable graph invariant for studying percolation phenomena.
The deletion-contraction algorithm provides an effective computational tool for its calculation.
The method yields highly accurate approximations for percolation thresholds, as evidenced by the kagome lattice results.