The linear Ising model and its analytic continuation, random walk

Abstract

A generalization of Gauss's principle is used to derive the error laws corresponding to Types II and VII distributions in Pearson's classification scheme. Student's $r$-pdf (Type II) governs the distribution of the internal energy of a uniform, linear chain, Ising model, while analytic continuation of the uniform exchange energy converts it into a Student $t$-density (Type VII) for the position of a random walk in a single spatial dimension. Higher dimensional spaces, corresponding to larger degrees of freedom and generalizations to multidimensional Student $r$- and $t$-densities, are obtained by considering independent and identically distributed random variables, having rotationally invariant densities, whose entropies are additive and generating functions are multiplicative.