98-764 van Enter A., Maes C., Schonmann R.H., Shlosman, S.
The Griffiths Singularity Random Field (561K, PostScript) Dec 22, 98
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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.

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