Urn Models in Physics and Genetics

Abstract

Urn models conceptualize general allocation problems in which we distribute, withdraw, and redistribute certain objects or units into a specified number of categories. We think of the categories as urns and the objects as balls. Depending on the specific urn model, the balls may be of different colors and distinguishable or indistinguishable. Urn models are special because they can be successfully used to model real phenomena in diverse areas such as physics, ecology, genetics, economics, clinical trials, modeling of networks, and many others. The aim is to understand the evolution of the content of the urns as distribution and redistribution according to some prespecified scheme progresses. There are many urn models in probability, and the allocation scheme depends on exactly which model one wishes to study. We introduce and provide basic information on some key urn models in this chapter. Classic references are Feller (1968) and Johnson and Kotz (1977). Bernoulli (1713) and Whitworth (1901) are two historically important monographs on urn models. More recent references include Gani (2004), Lange (2003), and Ivchenko and Medvedev (1997). Other specific references are given in the various sections of this chapter.