Quasilocality of projected Gibbs measures through analyticity techniques

We present two examples of projections of Gibbs measures. In the first example we prove high-temperature complete analyticity for the q-state Potts model. As a consequence we obtain complete analyticity also for the decimated Potts model. In the second example we prove quasilocality of the projections to the line of the pure phases of the two dimensional standard Ising system in the whole uniqueness region, and indicate why non-Gibbsianness can be expected to occur for higher dimensions.