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Polygons, Polyominoes and Polycubes pp 247–299Cite as

Limit Distributions and Scaling Functions

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Part of the book series: Lecture Notes in Physics ((LNP,volume 775))

For a given combinatorial class of objects, such as polygons or polyhedra, the most basic question concerns the number of objects of a given size (always assumed to be finite), or an asymptotic estimate thereof. Informally stated, in this overview we will analyse the refined question:

What does a typical object look like?

In contrast to the combinatorial question about the number of objects of a given size, the latter question is of a probabilistic nature. For counting parameters in addition to object size, one asks for their (asymptotic) probability law. To give this question a meaning, an underlying ensemble has to be specified. The simplest choice is the uniform ensemble, where each object of a given size occurs with equal probability.

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  1. Fakultät für Mathematik, Universität Bielefeld, Postfach 10 01 31, Bielefeld, 33501, Germany

    Christoph Richard

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  1. Department of Mathematics and Statistics, The University of Melbourne, Victoria, 3010, Australia

    Anthony J. Guttman

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Richard, C. (2009). Limit Distributions and Scaling Functions. In: Guttman, A.J. (eds) Polygons, Polyominoes and Polycubes. Lecture Notes in Physics, vol 775. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-9927-4_11

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