Random associations on a lattice

Abstract

Wishart and Hirschfeld have considered the following problem (1). Given n points in order on a line, suppose that they may be ‘black’ or ‘white’ independently with probability p and q = 1 − p. What is the probability distribution of the number of black-white joins? They give an exact solution to this problem and prove that it tends to normality as n increases. This result is of interest in several branches of science (Mood (2)). In the present note we consider the analogous problem in more than one dimension. This has applications in physics and agriculture (compare, Ising (3), Cochran (4) and Todd (5)). Another problem of similar type occurs when we consider the number of black points to be fixed and their arrangement to be random, but we do not consider this here. A problem of arrangements of similar type has been considered by Onsager.