Haller Karl, Kennedy Tom
Absence of renormalization group pathologies
near the critical temperature - two examples
(116K, TeX)
ABSTRACT. We consider real space renormalization
group transformations for Ising type systems which are formally defined
by
$$e^{-H^\prime(\s^\prime)} = \sum_\sigma T(\s,\s^\prime) e^{-H(\s)}
$$
where $T(\s,\s^\prime)$ is a probability kernel, i.e.,
$\sum_{\s^\prime} T(\s,\s^\prime)=1$, for every configuration $\s$.
For each choice of the block spin configuration $\s^\prime$, let
$\mu_{\s^\prime}$ be the measure on spin configurations $\s$ which
is formally given by taking the probability of $\s$ to be proportional to
$T(\s,\s^\prime) e^{-H(\s)}$.
We give a condition which is sufficient to imply that the
renormalized Hamiltonian $H^\prime$ is defined. Roughly speaking, the
condition is that the collection of measures $\mu_{\s^\prime}$ are
in the high temperature phase uniformly in the block spin configuration
$\s^\prime$. The proof of this result uses methods of Olivieri and Picco.
We use our theorem to prove that the first iteration of the renormalization
group transformation is defined in the following two examples:
decimation with spacing $b=2$ on the square lattice with
$\beta < 1.36 \beta_c$
and the Kadanoff transformation with parameter $p$
on the triangular lattice in a subset of the $\beta,p$ plane that
includes values of $\beta$ greater than $\beta_c$.