Methods for checking coupling from the past

Sometimes one wants a sample from an unknown distribution. We call a realization, without any errors, of a random variable with such a distribution, a perfect sample. If one have an ergodic nite Markov chain which has this wanted distribution as its unique stationary distribution, then with Propp-Wilson's method, coupling from the past, one can get such a perfect sample. It's attained by running the ergodic Markov chain, generated by randomly iterated functions, from a distant past into the present. A perfect sample is found if one in a realization encounters a composition of functions giving a constant value of the iterates from that point and backwards in time. Usually one doesn't give much notice to these constant functions, more than knowing that they exist and sooner or later will occur, and that when one has been encountered, one will get a perfect sample. The purpose of this thesis is to explore the possibilities of nding and collecting these constant functions and using them in order to, perhaps in a easier and less computationally-demanding way, obtaining perfect samples. We've chosen the Ising model to be the model under which we will study this idea of nding and collecting constant functions. The calculations and simulations have been done in MATLAB.