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λ > 4: An Improved Lower Bound on the Growth Constant of Polyominoes


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? > 4: An Improved Lower Bound on the Growth Constant of Polyominoes, illustrative photo

Credit: Radachynskyi Serhii

What is λ? The universal constant λ arises in the study of three completely unrelated fields: combinatorics, percolation, and branched polymers. In combinatorics, analysis of self-avoiding walks (SAWs, or non-self-intersecting lattice paths starting at the origin, counted by lattice units), simple polygons or self-avoiding polygons (SAPs, or closed SAWs, counted by either perimeter or area), and "polyominoes" (SAPs possibly with holes, or edge-connected sets of lattice squares, counted by area) are all related. In statistical physics, SAWs and SAPs play a significant role in percolation processes and in the collapse transition that branched polymers undergo when being heated. A collection edited by Guttmann15 gives an excellent review of all these topics and the connections between them. In this article, we describe our effort to prove that the growth constant (or asymptotic growth rate, also called the "connective constant") of polyominoes is strictly greater than 4. To this aim, we exploited to the maximum possible computer resources available to us. We eventually obtained a computer-generated proof that was verified by other programs implemented independently. We start with a brief introduction of the three research areas.

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Enumerative combinatorics. Imagine a two-dimensional grid of square cells. A polyomino is a connected set of cells, where connectivity is along the sides of cells but not through corners; see Figure 1 for examples of small polyominoes. Polyominoes were popularized in the pioneering book by Golomb13 and by Martin Gardner's columns in Scientific American; counting polyominoes by size became a popular and fascinating combinatorial problem. The size of a polyomino is the number of its cells.


 

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