218 research outputs found
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 steps has poles at
, where is a free parameter playing a role similar to the
dimensionality in nearest neighbor models.Comment: 24 Pages, includes 2 short Mathematica programs appended after "/end"
uses phyzzx.te
Universality, Scaling and Triviality in a Hierarchical Field Theory
Using polynomial truncations of the Fourier transform of the RG
transformation of Dyson's hierarchical model, we show that it is possible to
calculate very accurately the renormalized quantities in the symmetric phase.
Numerical results regarding the corrections to the scaling laws, (i.e finite
cut-off dependence) triviality, hyperscaling, universality and high-accuracy
determinations of the critical exponents are discussed.Comment: LATTICE98(spin
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