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This paper deals with the problem of perfect sampling from a Gibbs measurewith infinite range interactions. We present some sufficient conditions for theextinction of processes which are like supermartingales when large values aretaken. This result has deep consequences on perfect simulation, showing thatlocal modifications on the interactions of a model do not affect simulability.We also pose the question to optimize over a class of sequences of sets thatinfluence the sufficient condition for the perfect simulation of the Gibbsmeasure. We completely solve this question both for the long range Ising modelsand for the spin models with finite range interactions.

Developments in perfect simulation of Gibbs measures through a new result for the extinction of Galton-Watson-like processes