Cluster maximization, non-locality, and random tilings

Abstract

Jeong and Steinhardt (JS) implement local rules by selecting the sub-ensemble of tilings which have the maximum occurrences of a chosen pattern (``cluster'') C It is unknown how to prove that a given C implies a given sub-ensemble; counterexamples given here demonstrate this problem is nonlocal, and show the JS results depend on periodic boundary conditions. Sub-ensembles which are tilings of supertiles are at least as interesting, mathematically and physically, as the ideal cases emphasized by JS; the case C = star-decagon is explored here. Finally, I suggest minimizing the frequency of chosen intersite distances as a more physical alternative to the cluster approach.