Rigorous Bounds for Bond Percolation Thresholds of Three-dimensional Lattices
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Date
2018-10-18
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Johns Hopkins University
Abstract
In this dissertation we introduce and apply a new growth process methodology that provides rigorous upper bounds for the percolation thresholds of several three-dimensional bond percolation models. By studying rigorous upper and lower bounds of bond percolation thresholds, one hopes to obtain guidelines or hints for determining exact percolation thresholds. Meanwhile, rigorous upper and lower bounds may provide verifications or disproofs of the simulation estimates or conjectured values of the corresponding thresholds. A trustworthy result for the percolation threshold, whether exactly solved or obtained from simulations, provides an accurate phase transition point above which percolation (or infinite connectivity) occurs on the associated porous media, which is of great interest in industrial and scientific fields. Examples of applications include characterization of permeability, conductivity, diffusivity of porous media, biological evolution, spread of information, and many others.
The dissertation focuses on three-dimensional bond percolation models. Currently, none of the percolation thresholds of three-dimensional bond percolation models have been solved exactly. Moreover, there are few rigorous upper and lower bounds, and those that exist are not very precise. However, since most real-world media are three-dimensional, these models are of great importance in physics applications. This motivates us to study the percolation thresholds of such models.
To obtain upper bounds for the bond percolation thresholds of three-dimensional lattices, we develop a growth process approach. The growth process approach analyzes the static connected component in a random graph by considering it to be a dynamic process. The process describes the connected component as a growing set of vertices that may keep adding a subset of their current neighbors depending on the random states of the incident edges. Subsequently, we analyze whether or not this growth process stops growing to study whether percolation occurs on the associated lattice or not. \\
The methodology provides us with a powerful tool to study a variety of three-dimensional bond percolation models. The growth process approach works on a family of lattices called stacked lattices. It can also be slightly modified and applied to other well-studied three dimensional lattices such as the face-centered cubic lattice and the body-centered cubic lattice.
In this dissertation we provide detailed description, justification and applications of the growth process method. The results we obtain are quite satisfactory, compared with the previous rigorous bounds for our problem models. For instance, we prove that the bond percolation threshold of the cubic lattice, for which high-precision simulation estimates yield $0.24881$, is smaller than $0.34730$, in contrast to the previous best upper bound, $0.44779$. We thus successfully narrow the difference between the rigorous upper bound and simulation estimates for the percolation threshold of the cubic lattice bond percolation model by approximately $50\%$. However, there is still much room for further improvement. Future research may explore more approaches for bounding percolation thresholds, or even solving them; generalizing the growth process approach to a wider family of models (e.g., mixed percolation models and directed percolation models); and using our improved bounds as references and applying the substitution method to obtain bounds for other models.
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Keywords
Percolation threshold, Stochastic process