.1 Ising Model

In other words, full conditional distribution of Xn depends of only on the neighbors Xn−1 and Xn+1. In the 2-D setting, assume that S = {1, 2, . . . , N} × {1, 2, . . . , N} is the set of N2 points, called sites. For a fixed site s define a neighborhood N(s). For example for the site (i, j) the neighborhood could be N((i, j)) = {(i− 1, j), (i + 1, j), (i, j − 1), (i, j + 1)}, where one may take (0, j) = (N, j), (N + 1, j) = (1, j), (i, 0) = (i,N) and (i,N + 1) = (i, 1). Markov Property of X(S) is defined via local conditionals,