The High-Temperature Expansion of the Hierarchical Ising Model: From Poincar\'e Symmetry to an Algebraic Algorithm

We show that the hierarchical model at finite volume has a symmetry group which can be decomposed into rotations and translations as the familiar Poincar\'e groups. Using these symmetries, we show that the intricate sums appearing in the calculation of the high-temperature expansion of the magnetic susceptibility can be performed, at least up to the fourth order, using elementary algebraic manipulations which can be implemented with a computer. These symmetries appear more clearly if we use the 2-adic fractions to label the sites. We then apply the new algebraic methods to the calculation of quantities having a random walk interpretation. In particular, we show that the probability of returning at the starting point after $m$ steps has poles at $D=-2,-4,....-2m$ , where $D$ is a free parameter playing a role similar to the dimensionality in nearest neighbor models.