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Tight lower bound for percolation threshold on an infinite graph.

Kathleen E Hamilton¹, Leonid P Pryadko¹

¹Department of Physics and Astronomy, University of California, Riverside, California 92521, USA.

Physical Review Letters

|November 29, 2014

Summary

Researchers developed a new lower bound for site percolation thresholds on infinite graphs. This bound, derived from the Hashimoto matrix eigenvalue, is exact for infinite trees and improves upon existing methods.

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Area of Science:

Graph theory
Statistical mechanics
Percolation theory

Background:

Percolation thresholds are critical parameters in understanding connectivity in random networks.
Existing bounds for infinite graphs often rely on spectral radius or maximum degree, with limitations.

Purpose of the Study:

To establish a novel, tight lower bound for the site percolation threshold on infinite graphs.
To demonstrate the exactness of this bound for infinite trees.
To compare the new bound with existing methods regarding spectral radius and maximum degree.

Main Methods:

Constructing a lower bound using the maximal eigenvalue of the Hashimoto matrix.
Analyzing nonbacktracking walks on graphs.
Proving the existence of the eigenvalue for quasitransitive graphs.

Main Results:

A tight lower bound for the site percolation threshold was derived.
The bound is exact for infinite trees.
The new bound exceeds the inverse spectral radius and is generally tighter than degree-based bounds.

Conclusions:

The Hashimoto matrix eigenvalue provides a powerful tool for bounding percolation thresholds.
This method offers improved accuracy and applicability, particularly for infinite and quasitransitive graphs.