Fabio Martinelli , Enzo Olivieri
Some Remarks on Pathologies of Renormalization Group Transformations for the Ising Model
(31K, plain TeX)
ABSTRACT. The results recently obtained by van
Enter, Fernandez
and Sokal [EFS] on non-Gibbsianness of the measure
$\nu\,=\,T_b\,\mu_{\beta ,h}$ arising from the application
of a single decimation transformation $T_b$, with spacing
$b$, to the
Gibbs measure $\mu_{\beta ,h}$ of the Ising model, for
suitably
chosen large inverse temperature $\beta$ and non zero
external field
$h$, are critically analyzed. In particular we show that
if, keeping
fixed the same values of $\beta$, $h$ and $b$, one iterates
a
sufficiently large number of times $n$ the transfomation
$T_b$, one
obtains a new measure $\nu '\,=\,(T_b)^n\,\mu_{\beta
,h}$ which is
Gibbsian and moreover very weakly coupled.