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COMPUTATIONAL IMPROVEMENTS IN THE SUBSTITUTION METHOD FOR BOUNDING PERCOLATION THRESHOLDS
, 2005
"... The substitution method has proven to be an effective tool for bounding the critical probability of a variety of percolation models. Nevertheless, until recently substitution method calculations have been done by hand. This has severely restricted the size of computationally feasible substitution re ..."
Abstract

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The substitution method has proven to be an effective tool for bounding the critical probability of a variety of percolation models. Nevertheless, until recently substitution method calculations have been done by hand. This has severely restricted the size of computationally feasible substitution regions. We examine the computational problems posed by the substitution method, beginning with an analysis of the component calculations. We seek to better understand the nature of the computational problem, hoping that better understanding will lead to improvements. Our goal is a little different from that of most algorithmic investigations. Since the substitution method constitutes a proof, there is little reason to perform a particular computation more than once. We use each speed improvement to attempt a new, larger computation that will lead to tighter bounds on the critical probability. Our major results can be grouped into two categories: recognition of links between substitution method calculations and wellknown results in other areas of mathematics,