van Enter A., Maes C., Schonmann R.H., Shlosman, S.
The Griffiths Singularity Random Field
(561K, PostScript)
ABSTRACT. We consider a spin system on sites of a $d$-dimensional cubic lattice
($d\geq
2$), with the values $0,1$ or $-1$. It is built over the Bernoulli site
percolation model, with spins taking the value $0$ on empty sites, and
taking values $\pm 1$ on occupied sites according to the ferromagnetic
Ising
model distribution on the occupied clusters. The Hamiltonian corresponds
to
the nearest neighbor interaction under external field $h$, at inverse
temperature $\beta $, and the boundary conditions for clusters are free.
When the probability $p$ for a site to be occupied is small enough, so
that
a.s. all the clusters of non-$0$ spins are finite, this description gives
rise to a unique random field. We show that it is non-Markovian, and when
$p$
is small, $\beta $ is large and $h=0,$ it is even non-Gibbsian, but only
almost Gibbsian. This provides another example of a non-Gibbsian, but
almost
Gibbsian, random field which emerges naturally in a Gibbsian context. Our
random field is directly related to, and motivated by, the model studied
by
Griffiths in connection to what became known as the phenomenon of
Griffiths'
singularities.