One-dimensional infinite memory imitation models with noise

In this paper we study stochastic process indexed by $\mathbb {Z}$ constructed from certain transition kernels depending on the whole past. These kernels prescribe that, at any time, the current state is selected by looking only at a previous random instant. We characterize uniqueness in terms of simple concepts concerning families of stochastic matrices, generalizing the results previously obtained in De Santis and Piccioni (J. Stat. Phys., 150(6):1017--1029, 2013).Comment: 22 pages, 3 figure

Perfect simulation of autoregressive models with infinite memory

In this paper we consider the problem of determining the law of binary stochastic processes from transition kernels depending on the whole past. These kernels are linear in the past values of the process. They are allowed to assume values close to both 0 and 1, preventing the application of usual results on uniqueness. More precisely we give sufficient conditions for uniqueness and non-uniqueness. In the former case a perfect simulation algorithm is also given.Comment: 12 page

Convergence in Energy-Lowering (Disordered) Stochastic Spin Systems

We consider stochastic processes, S^t \equiv (S_x^t : x \in Z^d), with each S_x^t taking values in some fixed finite set, in which spin flips (i.e., changes of S_x^t) do not raise the energy. We extend earlier results of Nanda-Newman-Stein that each site x has almost surely only finitely many flips that strictly lower the energy and thus that in models without zero-energy flips there is convergence to an absorbing state. In particular, the assumption of finite mean energy density can be eliminated by constructing a percolation-theoretic Lyapunov function density as a substitute for the mean energy density. Our results apply to random energy functions with a translation-invariant distribution and to quite general (not necessarily Markovian) dynamics.Comment: 11 page