Gibbs Properties of the Bernoulli Field on Inhomogeneous Trees under the Removal of Isolated Sites

We consider the i.i.d. Bernoulli field mu p with occupation density p is an element of (0,1) on a possibly non -regular countably infinite tree with bounded degrees. For large p, we show that the quasilocal Gibbs property, i.e. compatibility with a suitable quasilocal specification, is lost under the deterministic transformation which removes all isolated ones and replaces them by zeros, while a quasilocal specification does exist at small p. Our results provide an example for an independent field in a spatially nonhomogeneous setup which loses the quasilocal Gibbs property under a local deterministic transformation.